mathematica vol. 6, no. 2, 2014

Acta Universitatis SapientiaeThe scientific journal of the Sapientia University publishes original papers and deep

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Main Editorial BoardZoltán A. BIRO Zoltán KASA András KELEMENAgnes PETHO Em˝od VERESS

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Sébastien FERENCZI (Institut de Mathématiques de Luminy, France)Kálmán GYORY (University of Debrecen, Hungary)

Zoltán MAKO (Sapientia University, Romania)Ladislav MISIK (University of Ostrava, Czech Republic)

János TOTH (Selye University, Slovakia)Adrian PETRUS¸EL (Babe¸s-Bolyai University, Romania)

Alexandru HORVATH (Petru Maior University of Tg.Mure¸s, Romania)Arpád BARICZ (Babe¸s-Bolyai University, Romania)Csaba SZANT O (Babe¸s-Bolyai University, Romania)Szilárd ANDRAS (Babe¸s-Bolyai University, Romania)

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Acta Universitatis Sapientiae

MathematicaVolume 6, Number 2, 2014

Sapientia Hungarian University of TransylvaniaScientia Publishing House

Contents

R. S. Batahan, A. A. BathanyaOn generalized Laguerre matrix polynomials . . . . . . . . . . . . . . . . . . 121

B. A. Bhayo, L. YinLogarithmic mean inequality for generalized trigonometric andhyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

A. A. Bouchentouf, H. SakhiStabilizing priority fluid queueing network model. . . . . . . . . . . . . 146

S. S. DragomirSome inequalities of Furuta’s type for functions of operators definedby power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

S. Mustonen, P. Haukkanen, J. MerikoskiSome polynomials associated with regular polygons . . . . . . . . . . . . 178

M. Z. Sarikaya, S. ErdenOn the weighted integral inequalities for convex function . . . . . . 194

N. Basu, A. BhattacharyyaEvolution of =-functional and ω-entropy functional for the conformalRicci flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Contents of volume 6, 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

119

Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 121–134

On generalized Laguerre matrixpolynomials

Raed S. BatahanDepartment of Mathematics,

Faculty of Science,Hadhramout University, 50511,

Mukalla, Yemenemail: [email protected]

A. A. BathanyaDepartment of Mathematics,

Faculty of Education,Shabwa, Aden University, Yemen

email: [email protected]

Abstract. The main object ofthe present paper is to introduce andstudy the generalized Laguerre matrix polynomials for a matrix thatsatisfies an appropriate spectralproperty.We prove that these matrixpolynomials are characterized by the generalized hypergeometric matrixfunction.An explicit representation,integralexpression and some re-currence relations in particular the three terms recurrence relation areobtained here. Moreover, these matrix polynomials appear as solution ofa differential equation.

1 IntroductionLaguerre, Hermite, Gegenbauer and Chebyshev matrix polynomials sequenceshave appeared in connection with the study ofmatrix differentialequations[8, 7, 20, 4]. In [13], the Laguerre and Hermite matrix polynomials were intro-duced as examples of right orthogonal matrix polynomial sequences for appro-priate right matrix moment functionals of integral type. The Laguerre matrixpolynomials were introduced and studied in [11, 14, 16, 17]. In [22], it is shownthat these matrix polynomials are orthogonal with respect to a non-diagonalSobolev-Laguerre matrix polynomials matrix moment functional. Recently, the2010 Mathematics Subject Classification: 15A15, 33C45, 42C05Key words and phrases: Laguerre matrix polynomials,three terms recurrence relation,generalized hypergeometric matrix function and Gamma matrix function

121

DOI: 10.1515/ausm-2015-0001

122 R. S. Batahan, A. A. Bathanya

numerical inversion of Laplace transforms using Laguerre matrix polynomialshas been given in [18]. A generalized form of the Gegenbauer matrix polyno-mials is presented in [2]. Moreover, two generalizations of the Hermite matrixpolynomials have been given in [1, 19].

The main aim of this paper is to consider a new generalization of the La-guerre matrix polynomials. The structure of this paper is the following. Aftera section introducing the notation and preliminary results, we characterize, inSection 3,the definition of the generalized Laguerre matrix polynomials andan explicit representation and integralexpression are given.Finally,Section4 deals with some recurrence relations in particular the three terms recur-rence relation for these matrix polynomials.Furthermore,we prove that thegeneralized Laguerre matrix polynomials satisfy a matrix differential equation.

2 PreliminariesThroughout this paper, for a matrix A in CN×N , its spectrum σ(A) denotes theset of all eigenvalues of A. We say that a matrix A is a positive stable if Re(µ)>0 for every eigenvalue µ ∈ σ(A). If f(z) and g(z) are holomorphic functions ofthe complex variable z, which are defined in an open set Ω of the complex planeand A is a matrix in CN×N with σ(A) ⊂ Ω, then from the properties of thematrix functional calculus [5, p. 558], it follows that f(A)g(A) = g(A)f(A). Thereciprocal gamma function denoted by Γ−1(z) = 1/Γ (z) is an entire function ofthe complex variable z. Then, for any matrix A in CN×N , the image of Γ−1(z)acting on A, denoted by Γ−1(A) is a well-defined matrix. Furthermore, if

A + nI is invertible for every integer n ≥ 0, (1)

where I is the identity matrix in CN×N , then Γ (A) is invertible,its inversecoincides with Γ−1(A) and it follows that [6, p. 253]

(A)n = A(A + I)...(A + (n − 1)I);n ≥ 1, (2)

with (A)0 = I.For any non-negative integers m and n, from (2), one easily obtains

(A)n+m = (A)n(A + nI)m, (3)

and(A)mn = mmn

mY

s=1

1m A + (s − 1)I

n. (4)

On generalized Laguerre matrix polynomials 123

Let P and Q be commuting matrices in CN×N such that for all integer n ≥ 0one satisfies the condition

P + nI, Q + nI, and P + Q + nI are invertible. (5)

Then by [10, Theorem 2] one gets

B(P, Q) = Γ (P)Γ (Q)Γ−1(P + Q), (6)

where the gamma matrix function, Γ (A), and the beta matrix function, B(P, Q),are defined respectively [9] by

Γ (A) =Z∞

0exp(−t)tA−Idt, (7)

andB(P, Q) =

Z1

0tP−I(1 − t)Q−Idt. (8)

In view of (7), we have [10, p. 206]

(A)n = Γ (A + nI)Γ−1(A); n ≥ 0. (9)

If λ is a complex number with Re(λ) > 0 and A is a matrix in CN×N withA + nI invertible for every integer n ≥ 1,then the n-th Laguerre matrixpolynomials L(A,λ)

n (x) is defined by [8, p. 58]

L(A,λ)n (x) =

nX

k=0

(−1)kλk

k!(n − k)!(A + I)n[(A + I)k]−1xk, (10)

and the generating function of these matrix polynomials is given [8] by

G(x, t, λ, A) = (1 − t)−(A+I)exp −λxt1 − t =

X

n≥0L(A,λ)

n (x)tn. (11)

According to [8],Laguerre matrix polynomials satisfy the three-term recur-rence relation

(n + 1)L(A,λ)n+1 (x) + λxI − (A + (2n + 1)I)L(A,λ)

n (x) + (12)(A + nI)L(A,λ)

n−1 (x) = θ; n ≥ 0,

with L(A,λ)−1 (x) = θ and L(A,λ)

0 (x) = I where θ is the zero matrix in CN×N .


Definition 1 [2]Let p and q be two non-negative integers.The generalizedhypergeometric matrix function is defined in the form:

pFq(A1, . . . , Ap; B1 . . . , Bq; z) (13)=

X

n≥0(A1)n . . . (Ap)n[(B1)n]−1. . . [(Bq)n]−1zn

n!,

where Ai and Bj are matrices in CN×N such that the matrices Bj; 1 ≤ j ≤ qsatisfy the condition (1).

With p = 1 and q = 0 in (13), one gets the following relation due to [11,p.213]

(1 − z)−A =X

n≥0

1n!(A)nzn, |z|< 1. (14)

The following lemma provides results about double matrix series. The proofare analogous to the corresponding for the scalar case c.f [15, p. 56] and [21,p. 101].Lemma 1 [2,3, 19]If A(k, n) and B(k, n) are matrices in CN×N for n ≥ 0and k ≥ 0, then it follows that:

X

n≥0

X

k≥0A(k, n) =

X

n≥0

nX

k=0A(k, n − k), (15)

X

n≥0

bn/mcX

k=0A(k, n) =

X

n≥0

X

k≥0A(k, n + mk), (16)

andX

n≥0

X

k≥0B(k, n) =

X

n≥0

bn/mcX

k=0B(k, n − mk) ; n > m, (17)

where bac is the standard floor function which maps a realnumber a to itsnext smallest integer.

It is obviously desirable, by (2), to have the following:1

(n − mk)!I = (−1)mk

n! (−nI)mk (18)

= (−1)mk

n! mmkmY

p=1

p − n − 1m I

k; 0 ≤ mk ≤ n.


3 Definition of generalized Laguerre matrix polyno-mials

Let A be a matrix in CN×N satisfying the spectral condition (1) and let λ bea complex number with Re(λ) > 0. For a positive integer m, we can define thegeneralized Laguerre matrix polynomials [GLMPs] by

F(x, t, λ, A) = (1 − t)−(A+I)exp

−λxmtm

(1 − t)m!

=∞X

n=0L(A,λ)

n,m (x)tn. (19)

By (14) one gets∞X

n=0

∞X

k=0

(−1)kλk

k!n! xmk(A + I + mkI)ntn+mk =∞X

n=0L(A,λ)

n,m (x)tn,

which by using (17) and (3) and equating the coefficients oftn, yields anexplicit representation for the GLMPs in the form:

L(A,λ)n,m (x) =

bn/mcX

k=0

(−1)kλk

k!(n − mk)!(A + I)n[(A + I)mk]−1xmk. (20)

It should be observed that when m = n,the explicit representation (20)becomes

L(A,λ)n,n (x) = (A + I)n

n! − λxnI.If m > n, then from (20) one gets

L(A,λ)n,m (x) = (A + I)n

n! .

Moreover, it is evident that

L(A,λ)n,m (0) =(A + I)n

n! and L(A,λ)n,m (x) = L(A,1)

n,m (λ 1m x).

Note that the expression (20) coincides with (10) for the case m = 1.In view of (4) and (18), we can rewrite the formula (20) in the form

L(A,λ)n,m (x) = (A + I)n

n!bn/mcX

k=0

(−1)(m+1)kλk

k! xmkmY

p=1

p − n − 1m I k

×" mY

s=1

1m(A + sI k

#−1.

(21)


Therefore, in view of (13), the hypergeometric matrix representation of GLMPsis given in the form:

L(A,λ)n,m (x) = (A + I)n

n!

mFm

−nm I, · · · ,(−n + m − 1)

m I; A + Im , · · · ,A + mI

m ; (−1)m+1λxm!

.(22)

We give a generating matrix function of GLMPs. This result is containedin the following.Theorem 1 Let A be a matrix in CN×N satisfying (1) and let λ be a complexnumber with Re(λ) > 0. Then

X

n≥0[(A + I)n]−1L(A,λ)

n,m (x)tn = et 0Fm

−;A + I

m , · · · ,A + mIm ; −λ xt

mm

!. (23)

Proof. By virtue of (20) and applying (16), we haveX

n≥0[(A + I)n]−1L(A,λ)

n,m (x)tn = X

n≥0

tn

n!

! X

k≥0

(−1)kλk

k! [(A + I)mk]−1xmktmk!

,

which, by using (4) and (13), reduces to (23).It is clear that

et 0Fm

−;A + I

m , · · · ,A + mIm ; −λ xyt

mm

!=

e(1−y)tety 0Fm

−;A + I

m , · · · ,A + mIm ; −λ xyt

mm

!.

Thus, by using (23) and applying (15), it follows thatX

n≥0[(A + I)n]−1L(A,λ)

n,m (xy)tn =X

n≥0

nX

k=0

(1 − y)n−kyk

(n − k)! [(A + I)k]−1L(A,λ)k,m (x)tn.

By equating the coefficients of tn, in the last series, one gets

L(A,λ)n,m (xy) = (A + I)n

nX

k=0

(1 − y)n−kyk

(n − k)! [(A + I)k]−1L(A,λ)k,m (x).


Let B be a matrix in CN×N satisfying (1). From (3), (4), (14) and (16) andtaking into account (20) we have

X

n≥0(B)n[(A + I)n]−1L(A,λ)

n,m (x)tn

= (1 − t)−B X

n≥0

(−λ)nn! (B)mn[(A + I)mn]−1 xt

1 − tmn

.(24)

By using (4) and (13), the equation (24) gives the following generating functionof GLMPs:

X

n≥0(B)n[(A + I)n]−1L(A,λ)

n,m (x)tn = (1 − t)−B

mFm

Bm, · · · ,B + (m − 1)I

m ;A + Im , · · · ,A + mI

m ; −λ xt1 − t

m!

.(25)

Clearly, (25) reduces to (19) when B = A + I.We now proceed to give an integral expression of GLMPs. For this purpose,

we state the following result.

Theorem 2 Let A and B be positive stable matrices in CN×N such that AB =BA. Then

L(A+B,λ)n,m (x) = Γ (A + B + (n + 1)I)Γ−1(B)Γ−1(A + (n + 1)I)

×Z1

0tA(1 − t)B−IL(A,λ)

n,m (xt)dt.(26)

Proof. According to (8) and (20), we can write

Ψ =Z1

0tA(1 − t)B−IL(A,λ)

n,m (xt)dt

=bn/mcX

k=0

(−1)kλk

k!(n − mk)!(A + I)n[(A + I)mk]−1xmkB(A + (mk + 1)I, B),(27)

and since the summation in the right-hand side of the above equality is finite,then the series and the integralcan be permuted.Hence by (6) and (9) it


follows that

Ψ = Γ A + (n + 1)IΓ (B)bn/mcX

k=0

(−1)kλkxmk

k!(n − mk)!Γ−1 A + B + (mk + 1)I

= Γ A + (n + 1)IΓ (B)Γ−1 A + B + (n + 1)Ibn/mcX

k=0

(−1)kλkxmk

k!(n − mk)!(A + B + I)n[(A + B + I)mk]−1.

(28)

From (20), (27) and (28), the expression (26) holds.We conclude this section giving an integral form of GLMPs.

Theorem 3 For GLMPs the following holdsZ∞

0xAL(A,λ)

n,m (x)e−xdx = Γ (A + (n + 1)I)n!

mF0

−nm , · · · ,−n + m − 1

m ; −; (−1)m+1λmm!

.(29)

Proof. From (7), (9) and (20), it follows that

Z∞

0xAL(A,λ)

n,m (x)e−xdx =bn/mcX

k=0

(−1)kλk

k!(n − mk)!(A + I)n[(A + I)mk]−1

Γ (A + (mk + 1)I)

= Γ (A + (n + 1)I)bn/mcX

k=0

(−1)kλk

k!(n − mk)!.

Using (18) and taking into account (13) we arrive at (29).

4 Recurrence relationsIn addition to the three terms recurrence relation, some differential recurrencerelations of GLMPs are obtained here.


Theorem 4 The generalized Laguerre matrix polynomials satisfy the follow-ing relations:

2mX

r=0

2mr (−1)r(n + 1 − r)L(A,λ)

n+1−r,m(x)

= (A + I)2m−1X

r=0

2m − 1r (−1)rL(A,λ)

n−r,m(x)

− mλxmmX

r=0

mr (−1)rL(A,λ)

n−r−m+1,m(x)

− mλxmm−1X

r=0

m − 1r (−1)rL(A,λ)

n−r−m,m(x),

(30)

and mX

r=0

mr (−1)rDL(A,λ)

n−r,m(x) = −λmxm−1L(A,λ)n−m,m(x). (31)

Proof. Differentiating (19) with respect to t yields

(1 − t)2mX

n≥1nL(A,λ)

n,m (x)tn−1 = (A + I)(1 − t)2m−1X

n≥0L(A,λ)

n,m (x)tn

−λmxmtm−1(1 − t)mX

n≥0L(A,λ)

n,m (x)tn − λxmtm(1 − t)m−1X

n≥0L(A,λ)

n,m (x)tn.

With the help of the binomial theorem, it follows thatX

n≥0

2mX

r=0

2mr (−1)r(n + 1)L(A,λ)

n+1,m(x)tn+r

= (A + I)X

n≥0

2m−1X

r=0

2m − 1r (−1)rL(A,λ)

n,m (x)tn+r

− λxm"m

X

n≥m−1

mX

r=0

mr (−1)rL(A,λ)

n−m+1,m(x)tn+r

+X

n≥m

m−1X

r=0

m − 1r (−1)rL(A,λ)

n−m,m(x)tn+r#.

Hence, by equating the coefficients of tn, equation (30) holds.


Now, by differentiating (19) with respect to x one getsX

n≥0DL(A,λ)

n,m (x)tn = −λmxm−1tm

(1 − t)m (1 − t)−(A+I)exp

−λxmtm

(1 − t)m!

.

Thus, it follows thatX

n≥m

mX

r=0

mr (−1)rDL(A,λ)

n−r,m(x)tn = −λmxm−1 X

n≥mL(A,λ)

n−m,m(x)tn,

which, by equating the coefficients of tn, gives (31).It is worthy to mention that (30) reduces to (12) for m = 1.Also, for

the case m = 1,the expression (31) gives the result for the Laguerre matrixpolynomials in the form

DL(A,λ)n (x) = DL(A,λ)

n−1 (x) − λL(A,λ)n−1 (x).

Differentiating (19) with respect to x again we obtainsX

n≥0DL(A,λ)

n,m (x)tn = −λmx m−1tm(1 − t)−(A+(m+1)I)exp

−λxmtm

(1 − t)m!

= −λmx m−1 X

n≥mL(A+mI,λ)

n−m,m (x)tn.

Hence, by equating the coefficients of tn, we readily obtain

DL(A,λ)n,m (x) = −λmxm−1L(A+mI,λ)

n−m,m (x). (32)It may be noted that the formula (32) reduces to the result of [12, p. 16] forLaguerre matrix polynomials, when m = 1, in the form

DL(A,λ)n (x) = −λL(A+I,λ)

n−1 (x).Using the fact that

(1 − t)−(A+I)exp −λxmtm

(1 − t)m = (1 − t)m(1 − t)−(A+(m+1)I)exp −λxmtm

(1 − t)m ,

and (19), one getsX

n≥0L(A,λ)

n,m (x)tn =X

n≥r

mX

r=0

mr (−1)rL(A+mI,λ)

n−r,m (x)tn.


Hence, we obtain that

L(A,λ)n,m (x) =

mX

r=0

mr (−1)rL(A+mI,λ)

n−r,m (x).

Let B be a matrix in CN×N satisfying (1) with AB = BA. Note that

(1 − t)−(A+I)exp

−λxmtm

(1 − t)m!

= (1 − t)−(A−B)(1 − t)−(B+I)exp

−λxmtm

(1 − t)m!

.

Using (14), (15) and (19) , it follows thatX

n≥0L(A,λ)

n,m (x)tn =X

n≥0

nX

k=0

(A − B)kk! L(B,λ)

n−k,m(x)tn.

Identifying the coefficients of tn, in the last series, gives

L(A,λ)n,m (x) =

nX

k=0

(A − B)kk! L(B,λ)

n−k,m(x). (33)

By reversing the order of summation in (33), we obtain that

L(A,λ)n,m (x) =

nX

k=0

(A − B)n−k(n − k)! L(B,λ)

k,m (x). (34)

And finally, we prove the following result.Theorem 5 The GLMPs is a solution of the following differentialequation

"Θ

mY

s=1

1m(Θ − 1)I +1

m(A + sI)!

+ (−1)mλmxm

×mY

p=1

1mΘ +p − n − 1

m

!I#L(A,λ)

n,m (x) = θ,(35)

where Θ = xddx.

Proof. It is clear that1mΘxmk = kxmk. According to (22) we can write

W = mFm − nmI, . . . , −n − m + 1

m I; A + Im , . . . ,A + mI

m ; (−1)m+1λ xm

=bn/mcX

k=0

mY

p=1

p − n − 1m k

"g

mY

s=1

1m(A + sI

k

#−1(−1)(m+1)kλk xmk

k! .


It follows after replacing k by k + 1 and using (3) that

1mΘ

mY

s=1

1m(Θ − 1)I +1

m(A + sI) W

=bn/mcX

k=0

mY

p=1

p − n − 1m k+1

" mY

s=1

1m(A + sI

k

#−1

(−1)(m+1)(k+1)λk+1 xm(k+1)

k!

= (−1)m+1λ xmbn/mcX

k=0

mY

p=1

p − n − 1m k+1

"g

mY

s=1

1m(A + sI

k

#−1

(−1)(m+1)kλk xmk

k!= (−1)m+1λ xm

mY

p=1

1mΘ +p − n − 1

m W.

Therefore, W is a solution of the following differential equation"

1mΘ

mY

s=1

1m(Θ−1)I+1

m(A+sI) +(−1)mλ xmmY

p=1

1mΘ+p − n − 1

m

#W = θ.

Since W = n![(A + I)n]−1L(A,λ)n,m (x), then (35) follows immediately.

It is worth noticing that taking m = 1 in (35) gives the following [8]"xI d2

dx2 + (A + (1 − λx)I)ddx + λnI#L(A,λ)

n (x) = θ.

AcknowledgmentsThe authors wish to express their gratitude to the unknown referee for severalhelpful suggestions.


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[11]L. Jódar, E. Defez, On Hermite matrix polynomials and Hermite matrixfunction, J. Approx. Theory Appl., 14 (1) (1998), 36–48.

[12]L. Jódar,E. Def ez, A Connection between Laguerre’s and Hermite’smatrix polynomials, Appl. Math. Lett., 11 (1) (1998), 13–17.

[13]L. Jódar,E. Defez,E. Ponsoda,Orthogonalmatrix polynomials withrespect to linear matrix moment functionals:Theory and applications,J. Approx. Theory Appl., 12 (1) (1996), 96–115.


[14]L. Jódar,J. Sastre,The growth of Laguerre matrix polynomialsonbounded intervals, Appl. Math. Lett., 13 (8) (2000), 21–26.

[15]E. D. Rainville, SpecialFunctions, The Macmillan Company, New York,(1960).

[16]J. Sastre,E. Defez,On the asymptotics ofLaguerre matrix polynomialfor large x and n, Appl. Math. Lett., 19 (2006), 721–727.

[17]J. Sastre,E. Defez,L. Jódar,Laguerre matrix polynomialseries expan-sion: Theory and computer applications,Math. Comput.Modelling,44(2006), 1025–1043.

[18]J. Sastre, E. Defez and L. Jódar, Application of Laguerre matrix polyno-mials to the numerical inversion of Laplace transforms of matrix functions,Appl. Math. Lett., 24 (9) (2011), 1527–1532.

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Received: 10 November 2014


Logarithmic mean inequality forgeneralized trigonometric and hyperbolic

functions

Barkat Ali BhayoDepartment of Mathematical

Information Technology, University ofJyväskylä, 40014 Jyväskylä, Finland


Li YinDepartment of Mathematics, BinzhouUniversity, Binzhou City, Shandong

Province, 256603, Chinaemail: yinli [email protected]

Abstract. In this paper we study the convexity and concavity prop-erties ofgeneralized trigonometric and hyperbolic functions in case ofLogarithmic mean.

1 IntroductionRecently, the study of the generalized trigonometric and generalized hyperbolicfunctions has got huge attention of numerous authors, and has appeared thehuge number of papers involving the equalities and inequalities and basis prop-erties of these function, e.g. see [7, 8, 9, 6, 10, 13, 14, 18, 23] and the referencestherein. These generalized trigonometric and generalized hyperbolic functionsp-functions depending on the parameter p > 1 were introduced by Lindqvist[19]in 1995.These functions coincides with the usualfunctions for p = 2.Thereafter Takesheu took one further step and generalized these function fortwo parameters p, q > 1, so-called (p, q)-functions. In [8], some convexity andconcavity properties of p-functions were studied. Thereafter those results wereextended in [5]for two parameters in the sense of Power mean inequality.Inthis paper we study the convexity and concavity property of p-function with2010 Mathematics Subject Classification: 33B10; 26D15; 26D99Key words and phrases:logarithmic mean,generalized trigonometric and hyperbolicfunctions, inequalities, generalized convexity

135

DOI: 10.1515/ausm-2015-0002

136 Generalized trigonometric and hyperbolic functions

respect Logarithmic mean. Before we formulate our main result we will definegeneralized trigonometric and hyperbolic functions customarily.

The eigenfunction sinp of the so-called one-dimensional p-Laplacian problem[12]

−∆pu = − |u0|p−2u0 0= λ|u|p−2u, u(0) = u(1) = 0, p > 1,

is the inverse function of F : (0, 1) → 0,πp2 , defined as

F(x) = arcsinp(x) =Zx

0(1 − tp)−1

p dt,

where

πp = 2arcsinp(1) = 2p

Z1

0(1 − s)−1/ps1/p−1ds = 2

p B 1 −1p, 1

p = 2πp sin π

p,

here B(., .) denotes the classical beta function.The function arcsinp is called the generalized inverse sine function,and

coincideswith usual inverse sine function forp = 2. Similarly,the othergeneralized inverse trigonometric and hyperbolic functions arccosp: (0, 1) →(0, πp/2) , arctanp: (0, 1) → (0, bp), arcsinhp: (0, 1) → (0, cp), arctanhp: (0, 1) →(0, ∞), where

bp = 12p ψ 1 + p

2p − ψ 12p = 2−1

p F 1p, 1

p; 1 +1p;1

2 ,

cp = 12

1p

F 1,1p; 1 +1p, 1

2 ,

are defined as follows

arccosp(x) =Z(1−xp )

1p

0(1 − tp)−1

p dt, arctanp(x) =Zx

0(1 + tp)−1dt,

arcsinhp(x) =Zx

0(1 + tp)−1

p dt, arctanhp(x) =Zx

0(1 − tp)−1dt,

where F(a, b; c; z) is Gaussian hypergeometric function [1].The generalized cosine function is defined by

ddx sinp(x) = cosp(x), x ∈ [0, πp/2] .

B. A. Bhayo, L. Yin 137

It follows from the definition thatcosp(x) = (1 − (sinp(x))p)1/p ,

and| cosp(x)|p + | sinp(x)|p = 1, x ∈ R. (1)

Clearly we getddx cosp(x) = − cosp(x)2−psinp(x)p−1.

The generalized tangent function tanp is defined by

tanp(x) = sinp(x)cosp(x),

and applying (1) we getddx tanp(x) = 1 + tanp(x)p.

For x ∈ (0, ∞), the inverse of generalized hyperbolic sine function sinhp(x)is defined by

arcsinhp(x) =Zx

0(1 + tp)−1/pdt,

and generalized hyperbolic cosine and tangent functions are defined by

coshp(x) = ddx sinhp(x), tanhp(x) = sinhp(x)

coshp(x) ,

respectively. It follows from the definitions that| coshp(x)|p − | sinhp(x)|p = 1. (2)

From above definition and (2) we get the following derivative formulas,ddx coshp(x) = coshp(x)2−psinhp(x)p−1, d

dx tanhp(x) = 1 − | tanhp(x)|p.

Note that these generalized trigonometric and hyperbolic functions coincidewith usual functions for p = 2.

For two distinct positive real numbers x and y, the Arithmetic mean, Geo-metric mean, Logarithmic mean, Harmonic mean and the Power mean of orderp ∈ R are respectively defined by

A(x, y) =x + y2 , G(x, y) =√ xy,


L(x, y) = x − ylog(x) − log(y), x 6= y,

H(x, y) = 1A(1/x, 1/y),

and

M t =

xt + yt

21/t

, t 6= 0,√ x y, t = 0 .

Let f : I → (0, ∞) be continuous,where I is a sub-intervalof (0, ∞). LetM and N be the means defined above, the we call that the function f is MN-convex (concave) if

f(M(x, y)) ≤ (≥)N(f(x), f(y)) for all x, y ∈ I .

Recently, Generalized convexity/concavity with respect to general mean val-ues has been studied by Anderson et al.in [2].We recallone of their resultsas follows

Lemma 1 [2,Theorem 2.4]Let I be an open sub-intervalof (0, ∞) and letf : I → (0, ∞) be differentiable.Then f is HH-convex (concave) on I ifandonly if x2f 0(x)/f(x)2 is increasing (decreasing).

In [4], Baricz studied that ifthe functions f is differentiable,then it is(a, b)-convex (concave) on I if and only if x1−af 0(x)/f(x)1−b is increasing (de-creasing).

It is important to mention that (1, 1)-convexity means the AA-convexity,(1, 0)-convexity means the AG-convexity, and (0, 0)-convexity means GG-convexity.

Motivated by the results given in [2, 4], we contribute to the topic by givingthe following result.

Theorem 1 Let f : I → (0, ∞) be a continuous and I ⊆ (0, ∞), then

1. L(f(x), f(y)) ≥ (≤)f(L(x, y)),2. L(f(x), f(y)) ≥ (≤)f(A(x, y)),

if f is increasing and log-convex (concave).

Theorem 2 For x, y ∈ (0, πp/2), the following inequalities

1. L(sinp(x), sinp(y)) ≤ sinp(L(x, y)), p > 1,


2. L(cosp(x), cosp(y)) ≤ cosp(L(x, y)), p ≥ 2.

Theorem 3 For p > 1, we have

1. L(1/ sinp(x), 1/ sinp(y)) ≥ 1/ sinp(A(x, y)), x, y ∈ (0, πp/2),

2. L(1/ cosp(x), 1/ cosp(y)) ≥ 1/ cosp(L(x, y)), x, y ∈ (0, πp/2),

3. L(tanhp(x), tanhp(y)) ≤ tanhp(A(x, y)), x, y ∈ (0, ∞),

4. L(arcsinhp(x), arcsinhp(y)) ≤ arcsinhp(A(x, y)), x, y ∈ (0, 1),

5. L(arctanp(x), arctanp(y)) ≤ arctanp(A(x, y)), x, y ∈ (0, 1).

2 Preliminaries and ProofsWe give the following lemmas which willbe used in the proofof our mainresult.

Lemma 2 [22]Let f, g : [a, b]→ R be integrable functions,both increasingor both decreasing.Furthermore,let p : [a, b]→ R be a positive,integrablefunction. Then

Zb

ap(x)f(x)dx

Zb

ap(x)g(x)dx ≤

Zb

ap(x)dx

Zb

ap(x)f(x)g(x)dx. (3)

If one of the functions f or g is non-increasing and the other non-decreasing,then the inequality in (3) is reversed.

Lemma 3 [17]If f(x) is continuous and convex function on [a, b],and ϕ(x)is continuous on [a, b], then

f 1b − a

Zb

aϕ(x)dx ≤ 1

b − aZb

af (ϕ(x)) dx. (4)

If function f(x) is continuous and concave on [a, b], then the inequality in (4)reverses.

Lemma 4 [3] For two distinct positive realnumbers a, b, we have L < A.

Lemma 5 For p > 1, the function sinp(x) is HH-concave on (0, πp/2).


Proof. Let f(x) = f 1(x)f2(x), x ∈ (0, πp/2), where f1(x) = 1/ sin(x) andf2(x) = x2cosp(x)/ sinp(x). Clearly,f1 is decreasing,so it is enough to provethat f2 is decreasing, then the proof follows from Lemma 1. We get

f 02(x) = sinp(x)(cosp(x) − x cosp(x)2−psinp(x)p−1) − x cosp(x)2

sinp(x)2

= cosp(x)2((1 − x tanp(x)p−1) tanp(x) − x)sinp(x)2 = f3(x)cosp(x)2

sinp(x)2 ,

where f3(x) = tanp(x) − x tanp(x)p − 1. Again, one hasf 03(x) = p tanp(x)p−1(1 + tanp(x)p)x < 0.

Thus, f3 is decreasing and g(x) < g(0) = 0.This implies that f02 < 0, hencef2 is strictly decreasing, the product of two decreasing functions is decreasing.This implies the proof.Proof of Theorem 1. We get

L(f(x), f(y)) =Rf(x)

f(y) 1dtRf(x)

f(y)1t dt

=Rx

y f 0(u)duRx

yf 0(u)f(u) du

. (5)

It is assumed that the function f(x) is increasing and log f is convex,thisimplies thatf 0(x)

f(x) is increasing.Letting p(x) = 1, f(x) = f(u) and g(x) =f 0(u)/f(u) in Lemma 2, we get

Zx

y1du

Zx

yf 0(u)du ≥

Zx

y

f 0(u)f(u) du

Zx

yf(u)du.

This is equivalent to

L(f(x), f(y)) =Rx

y f 0(u)duRx

yf 0(u)f(u) du

≥Rx

y f(u)duRxy 1du .

By Lemmas 3 and 4, and keeping in mind that log-convexity of f implies theconvexity of f, we get

L(f(x), f(y)) ≥ f Rx

y udux − y

!= f x + y

2 ≥ f (L(x, y)) .

The proof of converse follows similarly. If we repeat the lines of proof of part(1), and use the concavity of the function, and Lemmas 3 & 4 then we arriveat the proof of part (2).


Proof of Theorem 2. It is easy to see that the function sinp(x) is increasingand log-concave.So the proof of part (1) follows easily from Theorem 1.Wealso offer another proof as follows:

It can be observed easily that

L (sinp(x), sinp(y)) =Rx

y cosp(u)duRsinp (x)

sinp (y)1t dt

=Rx

y cospuduRxy

cospusinp (u)du,

andsinp (L (x, y)) = sinp

x − ylogx

y

!= sinp

Rxy 1du

Rxy

1udu

!.

Clearly, cosp(u) and sinp(1/u), utilizing Chebyshev inequality, we haveZx

ycosp(u)du

Zx

ysinp(1/u)du ≤

Zx

y1du

Zx

ycospusinp

1udu.

So, we getZx

ycospudu

Zx

ysinp(1/u)du <

Zx

y1du

Zx

y

cosp(u)sinp(u) du.

Where we apply simple inequality sinp 1u < 1

sinp (u) . In order to prove inequal-ity (1), we only prove

Rxy 1duRx

y sinp(1/u)du ≤ sinp Rx

y 1duRxy sinp(1/u)du

!.

Consider a partition T ofthe interval[y, x] into n equallength sub-intervalby means of points y = x0 < x1 < · · ·< xn = x and ∆xi = x−y

n . Picking anarbitrary point ξi ∈[xi−1, xi] and using Lemma 1.2, we have

nnP

i=1sinp 1

ξi

≤ sinp

nnP

i=11ξi

⇔x − y

limn→∞x−yn

nPi=1

sinp 1ξi

≤ sinp

x − ylimn→∞

x−yn

nPi=1

1ξi


⇔ Rxy 1duRx

y sinp(1/u)du ≤ sinp Rx

y 1duRxy sinp(1/u)du

!.

This completes the proof.For (2), clearly cosp(x) is decreasing and tanp(x)p−1 is increasing. One has

(cosp(x))00= cosp(x) tanp(x)p−2(1 − p +(2 − p) tanp(x)p) < 0,this implies that cosp(x) is concave on (0, πp/2).

Using Tchebyshef inequality, we haveZx

y1du

Zx

ycosp(u) tanp(u)p−1du ≤

Zx

ycosp(u)du

Zx

ytanp(u)p−1du,

which is equivalent toRx

y cosp(u) tanp(u)p−1duRxy tanp(u)p−1du ≤

Rxy cosp(u)duRx

y 1du . (6)

Substituting t = cosp(u) in (6), we get

L(cosp(x), cosp(y)) =Rcosp (x)

cosp (y) 1dtRcosp (x)

cosp (y)1t dt

=Rx

y cosp(u) tanp(u)p−1duRxy tanp(u)p−1du ≤

Rxy cosp(u)duRx

y 1du .

Using Lemma 3 and concavity of cosp(x), we obtain

L(cosp(x), cosp y) ≤ cosp Rx

y udux − y

!= cosp x + y

2 ≤ cosp (L(x, y)) .

Proof of Theorem 3.Let g1(x) = 1/ cosp(x), x ∈ (0, πp/2) and g2(x) =tanhp(x), x > 0. We get

(log(g1(x)))00= (p − 1) tanp(x)p−2(1 + tanp(x)p) > 0,and

(log(g2(x)))00= 1 − tanhp(x)ptanhp(x)2 ((1 − p) tanhp(x)p − 1) < 0.

This implies that g1 and g2 are log-convex, clearly both functions are increas-ing, and log-convexity implies the convexity, so g1 and g2 are convex functions.Now the proof follows easily from Theorem 1. The rest of proof follows simi-larly.


Corollary 1 For p > 1, we have

1. L(tanp(x), tanp(y)) ≥ tanp(L(x, y)), x, y ∈ (sp, πp/2), where sp is theunique root of the equation tanp(x) = 1/(p − 1)1/p,

2. L(arctanhp(x), arctanhp(y)) ≥ arctanhp(L(x, y)), x, y ∈ (rp, 1),whererp is the unique root of the equation xp−1arctanhp(y) = 1/p.

Proof. Write f1(x) = tanp(x). We get

f 01(x)f(x)

0= 1 + tanpp(x)

tanp(x)0= 1 + tanpp(x)

tan2p(x) (p − 1) tanpp(x) − 1 > 0

on sp, πp2 . This implies that f1 is log-convex, clearly f1 is increasing, and the

proof follows easily from Theorem 1. The proof of part (2) follows similarly.

AcknowledgementsThe second author was supported by NSF of Shandong Province under grantnumbers ZR2012AQ028, and by the Science Foundation of Binzhou Universityunder grant BZXYL1303.

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Stabilizing priority fluid queueing networkmodel

Amina AngelikaBouchentouf

Mathematics Laboratory, Djillali LiabesUniversity of Sidi Bel Abbes, B.P. 89,

Sidi Bel Abbes, Algeriaemail: bouchentouf−[email protected]

Hanane SakhiDepartment of Mathematics, Sciencesand Technology University of Oran:Mohamed Boudiaf, USTOMB, B.P.

1505 EL-M’NAOUAR- Oran, Algeriaemail: [email protected]

Abstract. The aim ofthis paper is to establish the stability offluidqueueing network models under priority service discipline.To this end,we introduce a priority fluid multiclass queueing network model,com-posed ofN stations,N ≥ 3 and 2N classes (2 classes at each station);where in the system, each station may serve more than one job class withdifferentiated service priority,and each job may require service sequen-tially by more than one service station.In this paper the fluid modelapproach is employed in the study of the stability.

1 IntroductionStochastic processing networks arise as models in manufacturing, telecommu-nications, computer systems and service industry. Common characteristics ofthese networks are that they have entities,such as jobs,customers or pack-ets,that move along routes,wait in buffers,receive processing from variousresources, and are subject to the effects of stochastic variability through suchquantities as arrival times, processing times, and routing protocols. Networksarising in modern applications are often highly complex and heterogeneous.Typically,their analysis and controlpresent challenging mathematicalprob-lems. One approach to these challenges is to consider approximate models.2010 Mathematics Subject Classification:60K25, 68M20, 90B22.Key words and phrases: stability,fluid models,multiclass queueing networks,fluid ap-proximation

146

DOI: 10.1515/ausm-2015-0003

Stabilizing priority fluid queueing network model 147

In the last 15 years, significant progress has been made on using approximatemodels to understand the stability and performance of a class of stochastic pro-cessing networks called open multiclass HL queueing networks. HL stands fora non-idling service discipline that is head-of-the-line, i.e., jobs are drawn froma buffer in the order in which they arrived.Examples of such disciplines areFIFO and static priorities. First order (functional laws of large numbers) ap-proximations called fluid models have been used to study the stability of thesenetworks, and second order (functional central limit theorem) approximationswhich are diffusion models,have been used to analyze the performance ofheavily congested networks.

The development of the fluid approach was inspired by the studies of somecounter-examples in Kumar and Seidman [11],Rybko and Stolyar [14]andBramson [1], etc., where the multiclass queueing networks are not stable evenwhen the traffic intensity of each station in the network is less than one.Anelegant result ofthe fluid modelapproach was proposed first in Rybko andStolyar [14]and then generalized and refined by Dai[6],Chen [2],Dai andMeyn [8],Stolyar [15]and Bramson [1].It states that a queueing network isstable if its corresponding fluid network model is stable. Partial converse to thisresult is also given in Meyn [12], Dai [7] and Puhalskii and Rybko [13]. HengQuing Ye [10]used Kumar-Rybko-Seidman-Stolyar network for establishingthe stability of fluid queueing network.

In this paper, we concentrate with the capacity of some large classes of fluidmulticlass queueing networks under priority service discipline. Specifically, weestablish a stability condition ofsome heterogenous priority fluid networkswith N stations and 2N job classes,where in the system,each station mayserve more than one job class with differentiated service priority,and eachjob may require service sequentially by more than one service station.So, inour case, the network performance is improved even when more workloads areadmitted for service.To stabilize our networks a number ofstations shouldbe added, these later act as regulators for the systems, adding these stationsis not random, it depends essentially on higher and lower priority job classes(many-to-one mapping) and on the number ofstations in the network.Thefluid model approach is employed to proof the stability.

The outline of the paper is as follows:At first (Section 2) we describe pri-ority fluid multiclass queueing models,and present a powerfulresult on thestability of such systems given by Chen and Zhang [5], after that (Section 3) weintroduce modified networks and present their stability conditions (Theorems2 and 3), and finally we conclude this paper with a short conclusion.

148 A. A. Bouchentouf, H. Sakhi

2 N-stations priority fluid multiclass network modelsWe describe N-stations priority fluid queueing network models as (J , K, λ, m,C, P, π). Specifically, the fluid network consists of J stations (buffers) (J = N)indexed by j ∈ J = 1, N, serving K, K = 2N fluid (customer) classes indexedby k ∈ K = 1, 2N.A fluid class is served exclusively at one station,but onestation may serve more than one fluid classes.σ(.) denotes a many-to-onemapping from K to J ,with σ(k) indicating the station at which a class kfluid is served. A class k fluid may flow exogenously into the network at ratesλ1 and λN+1, (≥ 0), then it is served at station σ(k), with mean service timemk = 1/µk, k = 1, 2N and after being served, a fraction pkl of fluid turns intoa class l fluid, l ∈ K, and the remaining fraction, 1 −P K

l=1pkl flows out of thenetwork.Let C(j) be the set of classes that reside in station j,alternatively,we denote by a J × K matrix C = (cij )J×K , known as the constituent matrix,where cjk = 1 if σ(k) = j, and cjk = 0 otherwise.

Let Qk(t) indicates the number of class k customers in the network at timet, (Q(0) = Qk(0)) and λ = (λk) two K-dimensional nonnegative vectors. P =(pkl)K×K a stochastic matrix with spectralradius strictly less that one,µ =(µk) a K-dimensional positive vector.

The vectors Q(0) are referred to as initial fluid level vector, λ to the exoge-nous inflow rate vector,µ to the processing rate vector,matrix P is referredto as flow transfer matrix.

When station σ(k) devotes its full capacity to serving class k fluid (assumingthat it is available to be served),it generates an outflow ofclass k fluid atrate µk > 0,k ∈ K. Among classes,fluid follows a priority service discipline,which is again described by a one-to-one mapping π from 1, ..., K onto itself.Specifically, a class k has priority over a class l if π(k) < π(l) and σ(l) = σ(k),then class k job can not be served at station σ(k) unless there is no class ljob.

So,our multiclass fluid network consists of N stations and 2N job classes.Assume that the arrival process of class k, k = 1, 2N, customers arrive to thesystem following a Poisson process with arrivalrates λ1 ≥ 0 and λN+1 ≥ 0,the service time for each class k customer is exponentially distributed withmean service time mk > 0. We also assume that all the inter-arrival times andservice times are independent.

To describe the dynamics of the fluid network, we introduce the K-dimensionalfluid levelprocess Q = Q(t),t ≥ 0,whose kth component Qk(t) denotesthe fluid level of class k at time t;the K-dimensional time allocation processT = T (t), t ≥ 0,whose kth component Tk(t) denotes the totalamount of


time that station σ(k) has devoted to serving class k fluid during the time in-terval [0, t], and the K-dimensional unused capacity process Y = Y(t), t ≥ 0,whose kth component Yk(t) denotes the (cumulative) unused capacity of sta-tion σ(k) during the time interval [0, t] after serving all classes at station σ(k)which have a priority no less than class k. We denote by D the K-dimensionaldiagonal matrix whose kth element is µk, and e is a K-dimensional vector withall components being one. Let

Hk = l : σ(l) = σ(k), π(l) ≤ π(k)

be the set of indices for all classes that are served at the same station as class kand have a priority no less than that of class k. Note that k ∈ Hk by definition.Then the dynamics of the fluid network model can be described as follows.

Q(t) = Q(0) + λt − (I − P0)DT (t) ≥ 0, (1)

T (·) are nondecreasing with T (0) = 0, (2)Yk(t) = t −

X

l∈Hk

Tl(t) are nondecreasing,k ∈ K, (3)

Z∞

0Qk(t)dYk(t) = 0, k ∈ K. (4)

Let

Qk(t) = Qk(0) + λkt +KX

l=1plkµlTl (t) − µkTk(t) ≥ 0, k = 1, . . . , K,(5)

be the kth coordinate of the flow balance relation (1).The equation (1) is the equivalent relation between the time allocation pro-

cess T (·) and the unused capacity process Y(.). The relation (4) means that atany time t, there could be some positive remaining capacity (rate) for servingthose classes at station σ(k) having a strictly lower priority than class k, onlywhen the fluid levels of all classes in Hk (having a priority no less than k) arezero.

A pair (Q, T ) (or equivalently (Q, Y)) is said to be a fluid solution if theyjointly satisfy (1)-(4).For convenience,we also callQ a fluid solution ifthere is a T such that the pair (Q, T ) is a fluid solution.The fluid network(J , K, λ, m, C, P, π) is said to be stable ifthere is a time τ ≥ 0 such thatQ(τ + ·) ≡ 0 for any fluid solution Q with kQ(0)k = 1;and it is said to weakly


stable if Q(·) = 0 for any fluid solution Q with Q(0) = 0. The processes Q, Y,and T are Lipschitz continuous, and hence are differentiable almost everywhereon [0, ∞), this well-known property will be used in this paper.

It is well-known that the queue length process Q(t) is a continuous timeMarkov chain under the Poisson arrival and exponential service assumptions.We say that the network (J , K, λ, m, C, P, π) is stable if the Markov chain Q(t)is positive recurrent.It is well-know that the Markov chain Q(t) is positiverecurrent only if the traffic intensity for each station is less than one, i.e., ρj < 1(ρj is the jth component of ρ;a traffic intensity for station j) for all j ∈ J , orin short, ρ < e, where e is a J-dimensional vector with all components beingones.

The expected stationary total queue length Q is defined as

Q = limt→∞ E" X

k∈KQk(t)

#.

The queue length Q(t) is a finite if and only if the queue length process Q ispositive recurrent.

Chen and Zhang [5]gave a very important result on the stability ofpri-ority fluid queueing systems,authors established the sufficient condition forthe stability based on the existence of a linear Lyapunov function,this later(sufficient condition) gave the the necessary and sufficient condition for thestability.Their result is presented in the Theorem 1,in order to state it weneed some additional assumptions:

Let

h(k) = arg maxπ(l) : l ∈ H+k if H+

k 6= ∅,0 otherwise, (6)

with H+k = Hk\k,in words;if k is not the highest priority class at station

σ(k), then h(k) is the index for the class which has the next higher prioritythan class k at station σ(k), otherwise h(k) = 0.

θ = λ − (I − P0)µ0H, (7)

where µ0H = De0H, (e0

H = (e01, ..., e0K)0) is a K-dimensional vector with e0

k = 1 ifH+

k = ∅ and e0k = 0 otherwise.

R = (I − P0)D(I − B), (8)


where B = (blk) is the K × K matrix with blk = 1 if k = h(l), and blk = 0otherwise, (l, k = 1, ...K).

And let

ρ = CD−1(I − P0)−1λ (9)be the traffic intensity of the queueing network.Theorem 1 [5]Consider a fluid network (λ, µ, P, C) under priority servicediscipline π. Let vector θ and matrix R be as defined in (7),(8) respectively. As-sume that ρ < e. Then the fluid network is stable if there exist a K-dimensionalvector h ≥ 0 such that for any given partition a and b of K satisfying if classl ∈ a, then each class k with

σ(k) = σ(l) and π(k) > π(l) is also in a, (10)

we haveh0

a(θa + Rabxb) < 0 (11)for xb ∈ Sb := u ≥ 0 :θb + Rbu = 0 and u ≤ e when b 6= ∅,and xb = 0when b = ∅. The inequality (11) is omitted to hold by default when Sb = ∅.

Set a includes allclasses which have zero unused capacity rate and set bincludes all classes which have a positive unused capacity rate at time t.

3 Main resultIn this paper,we present two theorems,we provide the proofof the firsttheorem, while the proof of the second one is omitted since it is similar to theformer one.

3.1 Stabilizing N-stations priority fluid queueing network withsome additional stations

Our multiclass queueing network consists ofN stations and 2N job classes.Assume that the arrival process of class k;k = 1, 2N customers arrive to thesystem following a Poisson process with arrivalrates λ1 and λN+1 (≥ 0) tostation 1 and N + 1 respectively,the service time for each class k customeris exponentially distributed with mean service time mk > 0. We also assumethat all the inter-arrival times and service times are independent.

Suppose that each even class at station i = 1, N has higher priority.


We modify our network such that ifit is composed ofan even number ofstations we add N additional ones otherwise we add (N − 1), the explanationof this choice willbe given in the rest ofthe paper,the modified networkis illustrated in Figures 1 and 2;the additionalstations are named stationN + 1,...,station 2N, (N: even) (resp. station N + 1,...,station 2N − 1 (N: odd)).

Figure 1:2N-stations priority fluid queueing network

Figure 2: 2N-1-stations priority fluid queueing network

Theorem 2 Suppose ρ < e, equation (11) not satisfied.If

λk1 > (1 − mk01/mk00

1)/mk1 (12)

then the queue length process Q(·) is positive recurrent.λk1 (resp. mk1) is the exogenous arrivalrate (resp. the mean service time)

of higher priority fluid class ofadditionalstations i = N + 1, 2N,(N: even)(resp.i = N + 1, 2N − 1,(N: odd)),such thatk1 = 3N + 1, 4N, (N :even)(resp.k1 = 3N, 4N − 2,(N : odd)). mk0

1is the mean service time oflower


priority fluid class of additionalstations i = N + 1, 2N,(N: even) (resp. i =N + 1, 2N − 1, (N: odd)). mk00

1 is the mean service time of higher priority fluidclass of the originalnetwork.

With

mk01

= mk1−N, k1 = 3N + 1, 4N,(N: even),mk1−(N−1), k1 = 3N, 4N − 2, (N: odd).

mk001 =

mk01−(2N−j1), k0

1 = 2N + 1,5N2 , j1 = 1,N2

(N: even),m(k1−k0

1)+j1, k01 = 5N+2

2 , k1 − N, j1 = 2, N

mk01−(2N−j1), k0

1 = 2N + 1,5N−12 , j1 = 1,N−1

2(N: odd).

m(k1−k01)+j1, k0

1 = 5N+12 , k1 − (N − 1), j1 = 4, N + 1

Where for each k01 it corresponds k1 and j1, (j1 is an even number).

Via this theorem, we show that when the arrival rates of some job classes isreduced, the performance of the queue will worse.Proof. In Chen and Yao [3] and Dai [7], it was shown that to prove the stabilityof a queueing model, it is sufficient to study the stability of its correspondingfluid queueing model, our prove is based on this result. To understand betterthe phenomenon,let us examine the dynamics ofthe originalnetwork withno initial job. When the higher priority job classes are being served, the lowerpriority ones are in standby,waiting for service,(class 1 jobs can not moveto class 2 and 2 cannot move to 3,... for further services, and vice versa). So,these classes will never be served at the same time and in effect form virtualstations (Dai and Vande Vate [9]). Therefore, the total nominal traffic intensityfor these classes together, i.e., the virtual stations, should not exceed one forthe network to be stable. The similar argument also yields that the network isunstable when the nominal traffic intensity for the virtual stations exceed one,i.e., the condition (11) is not satisfied. Now consider the modified network. Theadditional classes act as regulators that regulate the traffics in the network soas to stabilize the network.When the workloads of classes k1 (k1; defined inthe theorem) are light,much service capacity ofthe additionalstations areleft to classes k01 (k0

1; defined in the theorem) and hence these later do nothold back the traffics to avoid building up of job queues at priority classes of


the original network. Thus, the virtual stations effect prevails and the networkis still unstable when the condition (11) is not satisfied.However,when theworkloads of classes k1 are heavy enough such that the condition (12) holds,the service for lower priority classes at additionalstations is in effect sloweddown and the traffics in the original network are held back (these classes willnot mutually block their services). Finally, the virtual station effect is avoidedand the modified network is thus stabilized.

The dynamics of the our modified fluid network model can be described asfollows.

Qk1(t) = Qk1(0) + λk1t − µk1Tk1(t) ≥ 0, (13)k1 = 1, N + 1, 3N + 1, 4N, (N : even), (resp. k1 = 1, N + 1, 3N, 4N − 2)(N : odd),

Qk(t) = Qk(0) + µlTl(t) − µkTk(t) ≥ 0, (14)(k, l) = two successive job classes, such that the kth class is the arriving lth

classTk(·) are nondecreasing with Tk(0) = 0, (15)

k = 1, 4N, (N : even)(resp. k = 1, 4N − 2, (N : odd))

Yk1(t) = t − Tk1(t),are nondecreasing,

Yk001(t) = t − Tk00

1(t),

(16)

Yk(t) = t − Tl(t) − Tk(t) are nondecreasing, (17)(k, l) = (lower priority job class,higher priority job class ) at station i,i =1, 2N, (N : even) (resp. i = 1, 2N − 2,(N : odd)),

Z∞

0Qk(t)dYk(t) = 0, k = 1, 4N(N : even) (resp.k = 1, 4N − 2(N : odd)).

(18)The stability study of the modified fluid network will be done in three steps.

1. First step. We prove that there exists a time τ1 ≥ 0 such that

Qk1(t) = 0, for any t ≥ τ1, (19)with k1 = 3N + 1, 4N, (N: even), (resp. k1 = 3N, 4N − 2, (N: odd)).


If Qk1(t) > 0, then we have by equation (18)

Yk1(t) = 0, (20)then by conditions (16) and (20)

Tk1(t) = 1, (21)then by (13) and (21), we get

Qk1(t) = λk1 − µk1. (22)

Note that the condition ρ < e implies λk1 < µk1. Let τ(l)1 = Qk1(0)/(µk1 −

λk1), l = 1,N2 , (N: even) (resp. l = 1,N−12 , (N: odd)). Then, we have

Qk1(t) = 0 for any t ≥ τ(l)1 . (23)Letting τ1 = max(1/µk1 − λk1), we have that τ1 ≥ max(τ(l)1 ), (each l cor-

responds to k1) under the assumption kQ(0)k = 1.Now,the conclusion (23)leads to the claim (19).

2. Second step. We prove that there exists a time τ2 ≥ τ1 such that

Qk001(t) = 0, for any t ≥ τ2, (24)

where k001 is the higher priority job class at station i, i = 1, N.

k001 =

k01 − (2N − j1), k0

1 = 2N + 1,5N2 , j1 = 1,N2

(N: even),(k1 − k0

1) + j1, k01 = 5N+2

2 , k1 − N, j1 = 2, N

k01 − (2N − j1), k0

1 = 2N + 1,5N−12 , j1 = 1,N−1

2(N: odd).

(k1 − k01) + j1, k0

1 = 5N+12 , k1 − (N − 1), j1 = 4, N + 1

Under the condition (19), we haveQk1(t) = 0, and thenTk1(t) = λk1mk1, k1 =3N + 1, 4N, (N: even) (resp. k1 = 3N, 4N − 2, (N: odd)),for all time t ≥ τ1.Combined with (17), this gives rise to

Yk01(t) = t −Tk0

1(t) −Tk1(t) ≥ 0,


k01 are classes of lower priority at additional stations,

k01 = k1 − N, k1 = 3N + 1, 4N,(N: even),

k1 − (N − 1), k1 = 3N, 4N − 2,(N: odd).and

Tk01(t) ≤ 1 −Tk1(t) = 1 − λk1mk1, for any t ≥ τ1. (25)

Then,Qk00

1(t) = µk0

1Tk0

1(t) − µk00

1Tk00

1(t) ≤ µk0

1(1 − λk1mk1) − µk00

1< 0, where for

each k01 it corresponds k001 for any t ≥ τ2, where the last inequality is implied

by the assumption thatλk1 > (1 − mk0

1/mk00

1)/mk1.

Let τ(l)2 =

Qk 001

(τ1)µk 00

1−µk 0

1(1−λk1mk1) , l = 1, N (N: even),(resp.l = 1, N − 1 (N:

odd)). Then, we have

Qk001(t) = 0 for any t ≥ τ(l)2 . (26)

Letτ2 = max

1 + Θτ1

µk001

− µk01(1 − λk1mk1)

!

with Θ being the Lipschitz constant for the fluid level process Q(t). Then wehave that τ2 ≥ max(τ(l)2 ). Now, the conclusion (26) implies the claim (24).

Before to pass to the last step,we prove separately that QN+1(t) = 0 forany t ≥ τ2, “the case of network with even number of stations”.If QN+1(t) = 0, this impliesYN+1(t) = 0, which in turn implies thatTN+1(t) =1, then we haveQN+1(t) = λN+1 − µN+1, with λN+1 < µN+1 (since ρ < e). Sothere exists τ0

2 = QN+1(0)/µN+1−λN+1, such thatQN+1(t) = 0 for any t ≥ τ2.

Third step. We prove that there exists a time τ ≥ τ2 (≥ 0) such thatQl(t) = 0, for t ≥ τ, (27)

l represents job classes of lower priority at station i = 1, 2N (N: even) (resp.i = 1, 2N − 1 (N: odd), which together with equations (19) and (24) implies

Q(t) = 0 for t ≥ τ.


LetWi(t) = (λ1ml1 + λN+1ml2)t −

X

k:σ(k)=iTk(t), i = 1, N,

with l1 = 1, N and l2 = N = 1, 2N job classes at the same station in the orig-inal network.

Wi 0(t)=

λ1mk01t − Tk0

1(t), k0

1 = 2N + 1,5N2 , N: even,

(resp. k01 = 2N + 1, 2N +5N−1

2 , N: odd)λN+1mk0

1t − Tk0

1(t), k0

1 = 5N+22 , 3N, N: even,

(resp. k01 = 5N+1

2 , 3N − 1, N: odd)for τ ≥ τ2. Here W(t) can be explained as the immediate workload in thesystem at time t. Define

f i(t) = k01Wi(t), with k0

1 a lower priority job class in the additional stations.f i 0(t) = k00

1Wi 0(t), with k001 a higher priority job class in the original network.

For each i (resp. i0) it corresponds to k01 (resp. k00

1).Then, it is direct to verify that, for t ≥ τ2,

f i(t) < 0 if Qi(t) > 0, for i = 1, 4N, (N: even, (resp. i = 1, 4N − 2, (N: odd))And

f1(t) ≤ fN(t) if Q1(t) = 0,

fi(t) ≤ fi−1(t) if Qi(t) = 0, i = 2, 3N, (N: even)(resp. i = 2, 3N − 1, (N: odd))

fj(t) ≤ fi(t) ifX

j6=iQj(t) = 0, j = 1, 3N, (N: even)

(resp. j = 1, 3N − 1, (N: odd))fN(t) ≤ fN(t) if Q3N(t) = 0, (N: even) (resp. Q3N−1(t) = 0, (N: odd))

Now applying the piecewise linear Lyapunov function approach for the mul-ticlass fluid network model described in Theorem 3.1 of Chen and Ye [4], weobtain the conclusion (27).


3.2 Stabilizing N-stations priority fluid queueing networks withN additional stations

Our N-stations multiclass queueing network is the same is above. Suppose inthis case that the higher priority is devoted to classes N, N + 2, 2N.

Figure 3:2N-stations priority fluid queueing network

We modify our network by adding N stations,(see Figure 3),comparedto the originalnetwork,there are N additionalstations,namely the stationN + 1, . . . , station 2N, such that, 3N + 1 job class has high priority at stationN + 1, and 3N + 2,4N job classes have higher priority at stations N + 2, 2N.Now, let us introduce the second main result.

Theorem 3 Suppose ρ < e holds, equation (11) not satisfied.If

λ3N+1 > (1 − m2N+1/mN)/m3N+1, λk2 > (1 − mk02/mk00

2)/mk2, (28)

k002 = N, 2N, k0

2 = 2N + 1, 3N,k2 = 3N + 1, 4N,where for each k2 it corre-sponds to k02 and k00

2.Then the queue length process Q(·) is positive recurrent.

In this case,when the higher priority classes are being served,the lowerpriority ones cannot be served,(class 1 cannot move to class 2 and 2 cannotmove to 3,..., for further service, and vice versa. So, these later form a virtualstations.Therefore,these later,should not exceed one for the network to bestable. Now, let us consider the modified network. The additional classes 2N+1and 3N + 1, 4N act as regulators that regulate the traffics. When the workloadsof classes 3N + 1 and 3N + 2, 4N are light,much service capacity of stations


N + 1, . . . , 2N are left to classes 2N + 1, 3N respectively and hence these laterdo not hold back the traffics to avoid building up ofjob queues at higherpriority classes of the original network. Thus, the virtual stations effect prevailsand the network is stillunstable.However,when the workloads ofclasses3N + 1, 4N are heavy enough such that the condition (28) holds,the servicefor lower priority classes 2N + 1, 3N is in effect slowed down and the traffics tothe higher priority classes N and N + 2, 2N are held back. Finally, the virtualstations effect is avoided and the modified network is thus stabilized.

Then, following the same steps given in theorem 2, it is not difficult to provethat there exists a time τ1 ≥ 0 such that

Qk2(t) = 0, k2 = 3N + 1, 4N,for any t ≥ τ1. (29)after that, we prove that there exists a time τ2 ≥ τ1 such that

QN(t) = Qk002(t) = 0, k00

2 = N + 2, 2N for any t ≥ τ2. (30)and finally, we prove that there exists a time τ ≥ τ2(≥ 0) such that

Qk04(t) = 0 k0

4=lower priority job class at stations i = 1, 2N,for t ≥ τ.(31)

4 ConclusionMulticlass queueing networks are effective tools for modelling many indus-trial settings. One setting for which the model is particularly attractive is theproduction flow within semiconductor manufacturing facilities.

In this paper we have studied the stabilization of N-stations queueing net-works using its corresponding fluid network. The resulting model, fluid queue-ing networks with additionalstations depending on the service priority andon the number of stations in the network are formally presented in Section 3.Beyond the presentation of our modified network models “fluid networks withadditional stations”, the primary concern of the paper is the stability of suchnetworks.Nevertheless,stability of the artificialfluid modelimplies stabilityof the original network (see Theorems 2 and 3).

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sion of fluid limits, Queueing Syst., 23 (1998), 7–31.


[2]H. Chen, Fluid approximations and stability of multiclass queueing net-works: Workconserving discipline, Ann. Appl. Probab., 5 (1995), 637–655.

[3]H. Chen, D. D. Yao, Fundamentals of Queueing Networks: Performance,Asymptotics and Optimization, Springer-Verlag New York, Inc. (2001).

[4]H. Chen,H. Q. Ye, Piecewise linear Lyapunov function for the stabilityof priority multiclass queueing networks, IEEE Trans. Automat. Control,47 (4) (2002), 564–575.

[5]H. Chen, H. Zhang, Stability of multiclass queueing networks under pri-ority service disciplines, Oper. Res., 48 (2000), 26–37.

[6] J. G. Dai, On positive Harris recurrence of multiclass queueing networks:a unified approach via fluid models, Ann. Appl. Probab., 5 (1995), 49–77.

[7] J. G. Dai, A fluid-limit model criterion for instability of multiclass queue-ing networks, Ann. Appl. Probab., 6 (1996), 751–757.

[8] J. G. Dai, S. P. Meyn, Stability and convergence of moments for multiclassqueueing networks via fluid models,IEEE Trans. Automat. Control,40(1995), 1899–1904.

[9] J. G. Dai, J. H. Vande Vate,GlobalStability of Two-Station QueueingNetworks. Proceedings of Workshop on Stochastic Networks: Stability andRare Events,Editors:Paul Glasserman,Karl Sigman and David Yao,Springer-Verlag, Columbia University, New York., (1996), 1–26.

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Some inequalities of Furuta’s type forfunctions of operators defined by power

seriesSever S. Dragomir

Mathematics, College of Engineering & ScienceVictoria University, PO Box 14428

Melbourne City, MC 8001, AustraliaSchool of Computational & Applied Mathematics,

University of the Witwatersrand,Private Bag 3, Johannesburg 2050, South Africa


Abstract. Generalizations ofKato and Furuta inequalities for powerseries ofbounded linear operators in Hilbert spaces are given.Appli-cations for normaloperators and some functions of interest such as theexponential, hyperbolic and trigonometric functions are provided as well.

1 IntroductionIn the following we denote by B(H) the Banach algebra of all bounded linearoperators on a complex Hilbert space(H; h·, ·i) .

If P is a positive selfadjoint operator on H,i.e.hPx, xi ≥ 0 for any x ∈ H,then the following inequality is a generalization of the Schwarz inequality inH

|hPx, yi|2 ≤ hPx, xi hPy, yi , (1)for any x, y ∈ H.

The following inequality concerning the norm ofa positive operator is ofinterest as well, see [13, p. 221].2010 Mathematics Subject Classification: 47A63, 47A99Key words and phrases: Bounded linear operators, operator inequalities, Kato’s inequal-ity, functions of normal operators, Euclidian norm and numerical radius

162

DOI: 10.1515/ausm-2015-0004

Some inequalities of Furuta’s type 163

Let P be a positive selfadjoint operator on H. ThenkPxk2 ≤ kPk hPx, xi (2)

for any x ∈ H.The “square root” of a positive selfadjoint operator on H can be defined as

follows,see for instance [13,p. 240]:If the operator A ∈ B(H) is selfadjointand positive, then there exists a unique positive selfadjoint operator B :=

√A ∈

B(H) such thatB2 = A. If A is invertible, then so is B.If A ∈ B(H) , then the operator A∗A is selfadjoint and positive. Define the

“absolute value” operator by|A| :=√

A∗A.In 1952, Kato [14] proved the following celebrated generalization of Schwarz

inequality for any bounded linear operator T on H:

|hTx, yi|2 ≤D|T|2αx, x

E D|T∗|2(1−α) y, y

E(K)

for any x, y ∈ H and α ∈[0, 1].In order to generalize this result, in 1994 Furuta [12] obtained the following

result: DT |T|α+β−1x, y

E 2≤

D|T|2αx, x

E D|T∗|2β y, y

E(F)

for any x, y ∈ H and α, β ∈[0, 1] with α + β ≥ 1.If one analyses the proof from [12], that one realizes that the condition α, β

∈[0, 1] is taken only to fit with the result from the Heinz-Kato inequality

|hTx, yi|≤ kAαxk B1−αy (HK)

for any x, y ∈ H and α ∈[0, 1] where A and B are positive operators such thatkTxk ≤ kAxk and kT∗yk ≤ kByk for all x, y ∈ H.

Therefore, one can state the more general result:

Theorem 1 (Furuta Inequality, 1994, [12]) Let T ∈ B(H) and α, β ≥ 0with α + β ≥ 1. Then for any x, y ∈ H we have the inequality (F).

If we take β = α, then we getDT |T |2α−1x, y

E 2≤

D|T |2αx, x

E D|T∗|2αy, y

E(3)

for any x, y ∈ H and α ≥12. In particular, for α = 1 we get

|hT|T|x, yi|2 ≤D|T|2x, x

E D|T∗|2y, y

E(4)

164 S. S. Dragomir

for any x, y ∈ H.If we take T = N a normaloperator, i.e., we recall that NN∗= N∗N, then

we get from (F) the following inequality for normal operatorsDN |N|α+β−1x, y

E 2≤

D|N|2αx, x

E D|N|2β y, y

E(5)

for any x, y ∈ H and α, β ≥ 0 with α + β ≥ 1.This implies the inequalities

DN |N|2α−1x, y

E 2≤

D|N|2αx, x

E D|N|2αy, y

E(6)

for any x, y ∈ H and α ≥12 and, in particular,

|hN|N|x, yi|2 ≤D|N|2x, x

E D|N|2y, y

E(7)

for any x, y ∈ H.Making y = x in (6) produces

DN |N|2α−1x, x

E≤

D|N|2αx, x

E

for any x ∈ H and α ≥12 and, in particular,

|hN|N|x, xi| ≤D|N|2x, x

E

for any x ∈ H.If we take β = 1 − α with α ∈[0, 1] in (5), then we get

|hNx, yi|2 ≤D|N|2αx, x

E D|N|2(1−α) y, y

E(8)

for any x, y ∈ H.We can state the following corollary of Furuta’s inequality for the numerical

radius w of an operator V ∈ B(H), namely w(V) = supkxk=1|hVx, xi|, whichsatisfies the following basic inequalities

12kVk ≤ w (V) ≤ kVk .

Corollary 1 Let T ∈ B (H) and α, β ≥ 0 with α + β ≥ 1. Then we have

w T |T|α+β−1 ≤ 12 |T|2α +|T∗|2β . (9)


In particular, we also have

w T |T|2α−1 ≤ 12 |T|2α +|T∗|2α , (10)

for any α ≥12 and, as a specialcase,

w (T |T|)≤ 12 |T|2 +|T∗|2 . (11)

Proof. We have from (F) for any x ∈ H thatDT |T|α+β−1x, x

E≤

D|T|2αx, x

E1/2 D|T∗|2β x, x

E1/2(12)

≤ 12

Dh|T|2α +|T∗|2β

ix, x

E

where α, β ≥ 0 with α + β ≥ 1.Utilising the inequality in (12) and taking the supremum over x ∈ H, kxk = 1

we getw T |T|α+β−1 = sup

kxk=1

DT |T|α+β−1x, x

E

≤ 12 sup

kxk=1

Dh|T|2α +|T∗|2β

ix, x

E

= 12 |T|2α +|T∗|2β .

For various interesting generalizations,extension ofKato and Furuta in-equalities, see the papers [3]-[12], [17]-[21] and [23].

Motivated by the above results, we establish in this paper some generaliza-tions of Kato and Furuta inequalities for functions of operators that can beexpresses as power series with real coefficients. Applications for some functionsof interest such as the exponential, hyperbolic and trigonometric functions areprovided as well.

2 Functional inequalitiesNow, by the help of power series f(z) = P ∞

n=0anzn we can naturally constructanother power series which will have as coefficients the absolute values of thecoefficient of the originalseries,namely,fA (z) := P ∞

n=0|an|zn. It is obviousthat this new power series willhave the same radius ofconvergence as theoriginal series. We also notice that if all coefficients an ≥ 0, then fA = f.

166 S. S. Dragomir

Theorem 2 Let f(z) = P ∞n=0anzn and be g(z) = P ∞

n=0bnzn be two func-tions defined by power series with real coefficients and both of them convergenton the open disk D(0, R) ⊂ C, R > 0. If T is a bounded linear operator on theHilbert space H and z, u ∈ C with the property that

|z|2 , |u|2 , kT k2 < R, (13)

then we have the inequality

|hTf(z|T|)g(u |T|)x, yi|2 (14)≤ fA |z|2 gA |u|2

DfA |T|2 x, x

E D|T∗|2gA |T∗|2 y, y

E

for any x, y ∈ H.

Proof. From Furuta’s inequality (F) we have for any natural numbers n ≥ 0and m ≥ 1 the following power inequality

DT |T|n+m−1x, y

E≤

D|T|2nx, x

E1/2 D|T∗|2my, y

E1/2, (15)

where x, y ∈ H.If we multiply this inequality with the positive quantities|an| |z|n and|bm−1|

|u|m−1, use the triangle inequality and the Cauchy-Bunyakowsky-Schwarz dis-crete inequality we have successively:

kX

n=0

lX

m=1anznbm−1um−1

DT |T|n+m−1x, y

E(16)

≤kX

n=0

lX

m=1|an| |z|n |bm−1| |u|m−1

DT |T|n+m−1x, y

E

≤kX

n=0|an| |z|n

D|T|2n x, x

E1/2 lX

m=1|bm−1| |u|m−1

D|T∗|2my, y

E1/2

≤ kX

n=0|an| |z|2n

! 1/2 * kX

n=0|an| |T|2nx, x

+1/2

× lX

m=1|bm−1| |u|2(m−1)

! 1/2 * lX

m=1|bm−1| |T∗|2my, y

+1/2

for any x, y ∈ H and k ≥ 0, l ≥ 1.


Observe also thatkX

n=0

lX

m=1anznbm−1um−1

DT |T|n+m−1x, y

E(17)

=*

T kX

n=0anzn |T|n

! lX

m=1bm−1um−1|T|m−1

!x, y

+

for any x, y ∈ H and k ≥ 0, l ≥ 1.Making use of (16) and (17) we get

*T

kX

n=0anzn |T|n

! lX

m=1bm−1um−1|T|m−1

!x, y

+(18)

≤ kX

n=0|an| |z|2n

! 1/2 * kX

n=0|an| |T|2nx, x

+1/2

× lX

m=1|bm−1| |u|2(m−1)

! 1/2 *|T∗|2

lX

m=1|bm−1| |T∗|2(m−1) y, y

+1/2

for any x, y ∈ H and k ≥ 0, l ≥ 1.Due to the assumption (13) in the theorem, we have that the seriesP ∞

n=0anzn

|T|n , P ∞m=0bmum |T|m , P ∞

n=0|an| |T|2n andP ∞m=0|bm| |T∗|2m are convergent in

B(H) and the seriesP ∞n=0|an| |z|2n andP ∞

m=0|bm| |u|2m are convergent in Rand then, by taking the limit over k → ∞ and l → ∞ in (18), we deduce thedesired result (14).

Remark 1 The above inequality (14) can provide various particular instancesof interest.

For instance, if we take g = f in Theorem 2 then we getDTf2 (z|T|)x, y

E(19)

≤ fA |z|2 D

fA |T|2 x, xE1/2 D

|T∗|2 fA |T∗|2 y, yE1/2

for any x, y ∈ H.Also if we take g(z) = 1 in (14), then we get

|hTf(z|T|)x, yi|2 ≤ fA |z|2 D

fA |T|2 x, xE D

|T∗|2y, yE

(20)

for any x, y ∈ H.

168 S. S. Dragomir

Corollary 2 With the assumptions of Theorem 2 we have the norm inequality

kTf(z|T|)g(u |T|)k2 (21)≤ fA |z|2 gA |u|2 fA |T|2 |T∗|2gA |T∗|2

and the numericalradius inequality

w (Tf(z|T|)g(u |T|)) (22)

≤ 12

hfA |z|2 gA |u|2

i 1/2fA |T|2 +|T∗|2gA |T∗|2 .

Proof.The inequality (21) follows from (14) by taking the supremum overx, y ∈ H with kxk = kyk = 1.

From (14) we also have the inequality

|hTf(z|T|)g(u |T|)x, xi|

≤hfA |z|2 gA |u|2

i 1/2 DfA |T|2 x, x

E1/2 D|T∗|2gA |T∗|2 x, x

E1/2

≤ 12

hfA |z|2 gA |u|2

i 1/2 DhfA |T|2 +|T∗|2gA |T∗|2

ix, x

E1/2

for any x ∈ H,which,by taking the supremum over kxk = 1 produces thedesired result (22).

The following result also holds:

Theorem 3 Let f(z) = P ∞n=0anzn be a function defined by power series with

realcoefficients and convergent on the open disk D(0, R) ⊂ C, R > 0. If T is abounded linear operator on the Hilbert space H with the property that kT k2 < R,then we have the inequality

DT |T|f |T|2 x, y

E 2≤

D|T|2 fA |T|2 x, x

E D|T∗|2 fA |T∗|2 y, y

E(23)

for any x, y ∈ H.

Proof. From Furuta’s inequality (F) we have for any natural numbers n ≥ 1the power inequality

DT |T |2n−1x, y

E≤

D|T|2n x, x

E1/2 D|T∗|2n y, y

E1/2(24)

where x, y ∈ H.


If we multiply this inequality with the positive quantities|an−1|, use thetriangle inequality and the Cauchy-Bunyakowsky-Schwarz discrete inequalitywe have successively

* kX

n=1an−1T |T|2n−1x, y

+(25)

≤kX

n=1|an−1|

DT |T|2n−1x, y

E

≤kX

n=1|an−1|

D|T|2n x, x

E1/2 D|T∗|2n y, y

E1/2

≤* kX

n=1|an−1| |T|2n x, x

+1/2 * kX

n=1|an−1| |T∗|2n y, y

+1/2

for any x, y ∈ H and k ≥ 1.Observe also that

kX

n=1an−1T |T|2n−1= T|T|

kX

n=1an−1|T|2(n−1) ,

kX

n=1|an−1| |T|2n = |T|2

kX

n=1|an−1| |T|2(n−1)

andkX

n=1|an−1| |T∗|2n = |T∗|2

kX

n=1|an−1| |T∗|2(n−1)

for any k ≥ 1.Therefore, by (25) we have the inequality

*T |T|

kX

n=1an−1|T|2(n−1) x, y

+2

(26)

≤*

|T|2kX

n=1|an−1| |T|2(n−1) x, x

+ *|T∗|2

kX

n=1|an−1| |T∗|2(n−1) y, y

+

for any x, y ∈ H and k ≥ 1.

170 S. S. Dragomir

Due to the assumption kT k2 < R, we have that the seriesP ∞n=0an |T|2n ,P ∞

n=0|an| |T|2n and P ∞n=0|an| |T∗|2n are convergent in B(H) and taking the

limit over k → ∞ in (26) we deduce the desired result from (23).

Corollary 3 With the assumptions of Theorem 3 we have the norm inequality

T |T|f |T|22

≤ |T|2 fA |T|2 |T∗|2 fA |T∗|2

and the numericalradius inequality

w T |T|f |T|2 ≤ 12 |T|2 fA |T|2 +|T∗|2 fA |T∗|2 .

The following result for functions of normal operators holds.

Theorem 4 Let f(z) = P ∞n=0anzn be a function defined by power series with

realcoefficients and convergent on the open disk D (0, R) ⊂ C, R > 0.If N isa normaloperator on the Hilbert space H and α, β ≥ 0 with α + β ≥ 1 withthe property that kNk2α , kNk2β < R, then we have the inequality

Df N |N|(α+β−1) x, y

E 2≤

DfA |N|2α x, x

E DfA |N|2β y, y

E(27)

for any x, y ∈ H.

Proof. Utilising Furuta’s inequality written for Nn we haveDNn |Nn|α+β−1x, y

E 2≤

D|Nn|2αx, x

E D|(Nn)∗|2β y, y

E(28)

for any x, y ∈ H.Since N is normal, then

|Nn|2 = (Nn)∗Nn = N∗...N∗N...N= N ∗...NN∗...N = ...= (N∗N) ...(N∗N) = |N|2n

for any natural number n, and, similarly,

|(Nn)∗|2 = |(N∗)n|2 = |N∗|2n = |N|2n

for any n ∈ N.


These imply that |Nn|2α = |N|2αn, |(Nn)∗|2β = |N|2βn and |Nn|α+β−1 =|N|(α+β−1)n for any α, β ≥ 0 and for any n ∈ N.

Utilising the spectral representation for Borel functions of normal operatorson Hilbert spaces, see for instance [1, p. 67], we have for any α, β ≥ 0 and forany n ∈ N that

Nn |N|(α+β−1)n =Z

σ(N)zn |z|(α+β−1)ndP(z)

=Z

σ(N)

hz|z|(α+β−1)

i ndP(z)

=hN |N|(α+β−1)

i n,

where P is the spectral measure associated to the operator N and σ(N) is itsspectrum.

Therefore, the inequality (28) can be written asDh

N |N|(α+β−1)i n

x, yE

≤Dh

|N|2αi n

x, xE1/2 Dh

|N|2βi n

y, yE1/2

(29)for any x, y ∈ H and for any n ∈ N.

If we multiply the inequality (29) by|an| ≥ 0, sum over n from 0 to k ≥1 and utilize the Cauchy-Bunyakowsky-Schwarz discrete inequality,we havesuccessively

* kX

n=0an

hN |N|(α+β−1)

i nx, y

+(30)

≤kX

n=0|an|

DhN |N|(α+β−1)

i nx, y

E

≤kX

n=0|an|

Dh|N|2α

i nx, x

E1/2Dh|N|2β

i ny, y

E1/2

≤* kX

n=0|an|

h|N|2α

i nx, x

+1/2 * kX

n=0|an|

h|N|2β

i ny, y

+1/2

for any x, y ∈ H and for any k ≥ 1.Since kNk2α , kNk2β < R then N |N|(α+β−1) < R and the series

∞X

n=0|an|

h|N|2α

i n,

∞X

n=0|an|

h|N|2β

i n

172 S. S. Dragomir

and∞X

n=0an

hN |N|(α+β−1)

i n

are convergent in the Banach algebra B(H) .Taking the limit over k → ∞ in the inequality (30) we deduce the desired

result from (27).

Corollary 4 With the assumptions of Theorem 4, we have the inequality

f N |N|(α+β−1) 2≤ fA |N|2α fA |N|2β . (31)

Remark 2 If we take β = 1 − α with α ∈[0, 1] in (27), then we getthefollowing generalization of Kato’s inequality for normaloperators (8)

|hf(N)x, yi|2 ≤DfA |N|2α x, x

E DfA |N|2(1−α) y, y

E(32)

where x, y ∈ H and kNk2α , kNk2(1−α) < R.

3 ApplicationsAs some naturalexamples that are usefulfor applications,we can point outthat, if

f (z) =∞X

n=1

(−1)n

n! zn = ln 11 + z, z ∈ D(0, 1) ; (33)

g(z) =∞X

n=0

(−1)n

(2n) !z2n = cos z,z ∈ C;

h (z) =∞X

n=0

(−1)n

(2n + 1) !z2n+1= sin z,z ∈ C;

l (z) =∞X

n=0(−1)n zn = 1

1 + z, z ∈ D(0, 1) ;


then the corresponding functions constructed by the use of the absolute valuesof the coefficients are

fA (z) =∞X

n=1

1n!z

n = ln 11 − z, z ∈ D(0, 1) ; (34)

gA (z) =∞X

n=0

1(2n) !z

2n = cosh z,z ∈ C;

hA (z) =∞X

n=0

1(2n + 1) !z

2n+1= sinh z,z ∈ C;

lA (z) =∞X

n=0zn = 1

1 − z, z ∈ D(0, 1) .

Other important examples offunctions as power series representations withnonnegative coefficients are:

exp (z) =∞X

n=0

1n!z

n z ∈ C, (35)

12 ln 1 + z

1 − z =∞X

n=1

12n − 1z

2n−1, z ∈ D(0, 1) ;

sin−1(z) =∞X

n=0

Γ n + 12√ π (2n + 1) n!z

2n+1, z ∈ D(0, 1) ;

tanh−1(z) =∞X

n=1

12n − 1z

2n−1, z ∈ D (0, 1)

2F1(α, β, γ, z) =∞X

n=0

Γ (n + α) Γ (n + β) Γ (γ)n!Γ(α) Γ (β) Γ (n + γ) zn, α, β, γ > 0,

z ∈ D(0, 1) ;where Γ is the Gamma function.Example 1 Let x, y ∈ H.

a) If we take f(z) = sin z and g(z) = cos z in (14), then we get

|hT sin(z|T|)cos(u |T|)x, yi|2 (36)≤ sinh |z|2 cosh |u|2

×Dsinh |T|2 x, x

E D|T∗|2cosh |T∗|2 y, y

E

174 S. S. Dragomir

for any z ∈ C and T ∈ B(H) .b) If we take f(z) = ln 1

1+z and g(z) = ln 11−z in (14), then we get

DT ln(1H + z|T|)−1ln(1H − z|T|)−1x, y

E 2(37)

≤

ln 11 −|z|2

! 2

× ln 1H −|T|2−1

x, x

|T∗|2 ln 1H −|T∗|2−1

y, y

for any z ∈ C and T ∈ B(H) with|z| < 1 and kT k < 1.c) If we take f(z) = exp(z) and g(z) = exp(z) in (14), then we get

|hT exp[(z + u) |T|]x, yi|2 (38)≤ exp |z|2 exp |u|2

×Dexp |T|2 x, x

E D|T∗|2exp |T∗|2 y, y

E

for any z, u ∈ C and T ∈ B(H) .d) By the inequality (20) we have

DT sin−1(z|T|)x, y

E 2≤ sin−1 |z|2

Dsin−1 |T|2 x, x

E D|T∗|2y, y

E(39)

andDT tanh−1(z|T|)x, y

E 2(40)

≤ tanh−1 |z|2 D

tanh−1 |T |2 x, xE D

|T∗|2y, yE

for any z ∈ C and T ∈ B(H) with|z| < 1 and kT k < 1.

Example 2 Let x, y ∈ H.a) If we take f(z) = 1

1±z in (23), then we get

T |T| 1H ± |T|2−1

x, y2

(41)

≤ |T|2 1H −|T|2−1

x, x

|T∗|2 1H −|T∗|2−1

y, y


for any T ∈ B(H) with kT k < 1.b) If we take f(z) = ln 1

1±z in (23), then we get

T |T|ln 1H ± |T|2−1

x, y2

(42)

≤ |T|2 ln 1H −|T|2−1

x, x

|T∗|2 ln 1H −|T∗|2−1

y, y

for any T ∈ B(H) with kT k < 1.c) If we take f(z) = exp(z) in (23), then we get

DT |T|exp |T|2 x, y

E 2(43)

≤D|T|2exp |T|2 x, x

E D|T∗|2exp |T∗|2 y, y

E

for any T ∈ B(H) .Example 3 Let N be a normal operator on the Hilbert space H, α, β ≥ 0 withα + β ≥ 1 and x, y ∈ H.

a) If we take f(z) = 11±z in (27), then we get

1H ± N|N|(α+β−1) −1x, y

2(44)

≤ 1H −|N|2α −1x, x

1H −|N|2β −1

y, y

provided kNk < 1.In particular, we have

D(1H ± N)−1x, y

E 2(45)

≤ 1H −|N|2α −1x, x

1H −|N|2(1−α) −1

y, y ,

for α ∈[0, 1].b) If we take f(z) = exp(z) in (27), then we getDexp N |N|(α+β−1) x, y

E 2≤

Dexp |N|2α x, x

E Dexp |N|2β y, y

E. (46)

As a specialcase, we have

|hexp (N) x, yi|2 ≤Dexp |N|2α x, x

E Dexp |N|2(1−α) y, y

E, (47)

for α ∈[0, 1].

176 S. S. Dragomir

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Received: 13 June 2014


Some polynomials associated with regularpolygons

Seppo MustonenDepartment of Mathematics and

StatisticsFI-00014 University of Helsinki, Finland


Pentti HaukkanenSchool of Information Sciences

FI-33014 University of Tampere,Finland


Jorma MerikoskiSchool of Information Sciences

FI-33014 University of Tampere, Finlandemail: [email protected]

Abstract. Let Gn be a regular n-gon with unit circumradius, and m =bn

2 c, µ = bn−12 c. Let the edges and diagonals of Gn be en1 < · · · < enm .

We compute the coefficients of the polynomial

(x − e2n1 ) · · · (x − e2nµ ).

They appear to form a well-known integer sequence, and we study certainrelated sequences, too. We also compute the coefficients of the polynomial

(x − s2n1 ) · · · (x − s2nm ),

wheresni = cot(2i − 1)π

2n , i = 1, . . . , m.

We interpret sn1 as the sum of all individual edges and diagonals of Gndivided by n.We also discuss the interpretation ofsn2 , . . . , snm , andpresent a conjecture on expressing sn1 , . . . , snm using en1 , . . . , enm .

2010 Mathematics Subject Classification: 11B83, 11C08, 15B36, 51M20Key words and phrases: polynomial, regular polygon, eigenvalue

178

DOI: 10.1515/ausm-2015-0005

Polynomials associated with regular polygons 179

1 IntroductionThroughout, Gn is a regular n-gon with unit circumradius, and

m =j n

2k

, µ = n − 12 .

Long time ago Kepler observed [2] that the squares of the edge and diagonalsof G7 are the zeros of the polynomial x3− 7x2+ 14x − 7. This raises a generalquestion:Are the squares of(the lengths of) the edge and diagonals ofGn,excluding the diameter,the zeros ofa monic polynomialof degree µ withinteger coefficients?

Yes, they are.This follows from Savio’s and Suruyanarayan’s [6]results,which,however,do not give the polynomialexplicitly.We will do it in Sec-tion 2. A natural further question concerns the edge and diagonals themselves,instead of their squares. They are not zeros of a polynomial described above,but we will in Section 3 see that the squared sum of all individual edges anddiagonals is the largest zero ofa monic polynomialof degree m with inte-ger coefficients.We will study geometric interpretation ofthe square rootsof the other zeros in Section 4.In Section 5,we willpresent a conjecture onexpressing these square roots as simple linear combinations of the edge and di-agonals. We will in Section 6 notify that the coefficients of the first-mentionedpolynomial form an OEIS [4] sequence, and also study OEIS sequences corre-sponding to certain related polynomials.Finally,we willcomplete our paperwith conclusions and further questions in Section 7.

2 Squared chordsLet (the lengths of) the edge and diagonals of Gn be en1 < · · ·< enm. Callthem (the lengths of) the chords. Then

enk = 2 sinkπn , k = 1, . . . , m.

Our problem is to find the coefficients amk and bmk of the polynomialsAm(x) = (x − e2n+2,1) · · · (x − e2

n+2,m) =xm + am,m−1xm−1+ · · · + am1x + am0, (1)

where n is even, andBm(x) = (x − e2n1) · · · (x − e2

nm) = xm + bm,m−1xm−1+ · · · + bm1x + bm0, (2)

180 S. Mustonen, P. Haukkanen, J. Merikoski

where n is odd. We solve it in two theorems. Mustonen [3] found them exper-imentally and sketched their proofs.

Let tridiagm(x, y) denote the symmetric tridiagonal m × m matrix with allmain diagonal entries x and first super- and subdiagonal entries y. For m ≥ 2,define

Am = tridiagm(2, 1)and

Bm is as Am but the (m, m) entry equals 3.Also define A1 = (2) and B1 = (3). Denote by spec the (multi)set of eigenval-ues.

Lemma 1 For allm ≥ 1,

spec Am = 4 sin2 kπn + 2 k = 1, . . . , m= e2n+2,1, . . . , e2

n+2,m, (3)

spec Bm = 4 sin2 kπn k = 1, . . . , m= e2n1, . . . , e2

nm.

Proof. See [1, 5, 6].

Theorem 1 In (1),

amk = (−1)m−k m + 1 + k2k + 1 . (4)

Proof. Denoting

Pm(x) = xm +m−1X

k=0(−1)m−k m + 1 + k

2k + 1 xk,

our claim is that

Pm(x) = Am(x) (5)

for all m ≥ 1. Expanding det (xIm − Am) along the last row, we have

Am+1(x) = (x − 2)Am(x) − Am−1(x)

for all m ≥ 2. SinceP1(x) = x − 2 = A1(x)


andP2(x) = x2 − 4x + 3 = A2(x),

the claim (5) follows by showing that

Pm+1(x) = (x − 2)Pm(x) − Pm−1(x) (6)

for all m ≥ 2. Mustonen [3]did it by using Mathematica.We will do thecomputations algebraically in the appendix.

The formula (4) yields amm = 1, consistently with the coefficient ofxm

in (1). It also allows to define a00= 1. The polynomial

Am+1(x) = (x − 4)Am(x) = xm+1+ αm+1,mxm + · · · + αm+1,1x + αm+1,0 (7)

has e2n+2,m+1= 4 as the additional zero. By (4),

αm+1,k= (−1)m−k+1 m + k2k − 1 + 4 m + 1 + k

2k + 1 . (8)

(We definenk = 0 if k < 0.)

Theorem 2 In (2),

bmk = (−1)m−k2m + 1m − k

m + k2k + 1 = (−1)m−k m + 1 + k

2k + 1 + m + k2k + 1 . (9)

Proof.The second equation follows from trivialcomputation.To show thefirst, denote

Qm(x) = xm +m−1X

k=0(−1)m−k2m + 1

m − km + k2k + 1 xk

and claim that

Qm(x) = Bm(x) (10)

for all m ≥ 1. Expanding det (xIm − Bm), we have

Bm+1(x) = (x − 3)Am(x) − Am−1(x)

for all m ≥ 2. SinceQ1(x) = x − 3 = B1(x)


andQ2(x) = x2 − 5x + 5 = B2(x),

the claim (10) follows by showing that

Qm+1(x) = (x − 3)Pm(x) − Pm−1(x) (11)

for all m ≥ 2. Mustonen [3] did also this by using Mathematica, and we willdo the computations algebraically in the appendix.

For k = m, the first expression in (9) is undefined but the second is defined.(We definen

k = 0 if n < k.) It gives bmm = 1, the coefficient of xm in (2). Italso allows to define b00= 1.

Corollary 1 The sum ofall individualsquared chords ofGn is n2. Theirproduct is nn.

Proof. By Theorems 1 and 2 (or by [7, Eqs. (20) and (24)]), we obtain

e22m,1+ · · · + e2

2m,m−1= −am−1,m−2= 2(m − 1),e2

2m+1,1+ · · · + e22m+1,m= −bm,m−1= 2m + 1,

and

e22m,1· · · e22m,m−1= (−1)mam−1,0= m,

e22m+1,1· · · e22m+1,m= (−1)mbm0 = 2m + 1.

Denoting by Σn the sum and by Πn the product ofall individualsquaredchords of Gn, we therefore have

Σ2m = 2m · 2(m − 1) + m · 4 = (2m)2,Σ2m+1= (2m + 1)(2m + 1) = (2m + 1)2,

and

Π2m = m2m4m = (2m)2m, Π2m+1= (2m + 1)2m+1.


3 Sum of chordsThe sum of all individual chords of Gn is

Sn = nsn,

wheresn = en1 + · · · + en,m−1+ 1

2enm = en1 + · · · + en,m−1+ 1if n is even, and

sn = en1 + · · · + enm

if n is odd, is the sum ofdifferent (lengths of) chords but the diameter ishalved.

Theorem 3 For alln ≥ 3,sn = cot π

2n.

Proof. We have [7, Eq. (21)]n−1X

k=1sinkπ

n = cot π2n. (12)

If n is even, this implies

sn =m−1X

k=12 sinkπ

n + 12 · 2 =

m−1X

k=1sinkπ

n + 1 +2m−1X

k=m+1sinkπ

n =

2m−1X

k=1sinkπ

n = cot π2n.

If n is odd, then

sn =mX

k=12 sinkπ

n =mX

k=1sinkπ

n +2mX

k=m+1sinkπ

n =2mX

k=1sinkπ

n = cot π2n.

Is sn a zero of a monic polynomial of degree m with integer coefficients? Yesfor s4 = cotπ8 = 1+√ 2; it is a zero of x2− 2x − 1. On the other hand, it is easyto see that s5 = cot π

10 =p

5 + 2√

5 is not a zero of such a polynomial.But


s25 = 5 + 2

√5 is a zero of x2− 10x + 5, and the other zero is 5 − 2

√5 = cot2 3π

10.Also s24 = 3 + 2

√2 has this property: it is a zero of x2 − 6x + 1, and the other

zero is 3 − 2√

2 = cot2 3π8 .

Generally, denoting

sni = cot(2i − 1)π2n , i = 1, . . . , m,

this motivates us to study for even n the coefficients of the polynomial

Um(x) = (x − s2n1) · · · (x − s2nm) = xm + um,m−1xm−1+ · · · + um1x + um0, (13)

and for odd n those of

Vm(x) = (x − s2n1) · · · (x − s2nm) = xm + vm,m−1xm−1+ · · · + vm1x + vm0. (14)

We will see that they all are integers. The largest zero is s2n = s2n1.Mustonen [3] found the following theorem experimentally and also presented

its proof. Yaglom and Yaglom [9, Eqs. (7) and (8)] formulated (16) differently.

Theorem 4 In (13),

umk = (−1)k n2k . (15)

In (14),

vmk = (−1)k n2k + 1 . (16)

Proof. We have [10]

cot nt =P m

k=0(−1)k n2k cotn−2ktP m

k=0(−1)k n2k+1 cotn−2k−1t . (17)

Denoteti = (2i − 1)π

2n , i = 1, . . . , m.Since cot nti = 0, (17) yields

mX

k=0(−1)k n

2k cotn−2kti = 0. (18)


First assume n even. The polynomial

Um(x) =mX

k=0(−1)m−k n

2k xk

is monic and has degree m. For all i = 1, . . . , m,

Um(s2ni ) =

mX

k=0(−1)m−k 2m

2k s2kni =

mX

l=0(−1)l 2m

2m − 2l s2m−2lni

=mX

l=0(−1)l 2m

2l s2m−2lni =

mX

l=0(−1)l n

2l cotn−2l ti = 0

by (18). Hence

Um(x) = (x − s2n1) · · · (x − s2nm) = Um(x),

and (15) follows.Second, assume n odd. The polynomial

Vm(x) =mX

k=0(−1)m−k n

2k + 1 xk

is monic and has degree m. For all i = 1, . . . , m,

Vm(s2ni ) =

mX

k=0(−1)m−k 2m + 1

2k + 1 s2kni =

mX

l=0(−1)l 2m + 1

2m − 2l + 1s2m−2lni =

s−1ni

mX

l=0(−1)l 2m + 1

2m − 2l + 1s2m+1−2lni = s−1

ni

mX

l=0(−1)l 2m + 1

2l s2m+1−2lni

= s−1ni

mX

l=0(−1)l n

2l cotn−2l ti = 0,

again by (18). Hence

Vm(x) = (x − s2n1) · · · (x − s2nm) = Vm(x),

and (16) follows.


Corollary 2 The number s2n is the largest zero of the polynomial

xm + um,m−1nxm−1+ · · · + um1nm−1x + um0nm

if n is even, and that of

xm + vm,m−1nxm−1+ · · · + vm1nm−1x + vm0nm

if n is odd.

4 Interpreting sn,m−k+1, k = 1, . . . , bn−13 c, n odd

The zeros ofAm(x) and Bm(x) describe the squared chords ofG2m+2 andG2m+1, respectively, excluding the diameter. The largest zero of Um(x), s22m,1=s22m, and that of Vm(x), s2

2m+1,1= s22m+1, describe the squared sum of chords

but halving the diameter.In other words,the sum ofall individualchordsof Gn is divided by n and the result is squared.

What about the other zeros?Let the vertices ofGn be P0, . . . , Pn−1, where Pk = (coskπ

n , sinkπn ). Then

enk = P0Pk = 2 sinkπn , k = 1, . . . , m. Since P0Pn−k = P0Pk, we define en,n−k =

enk, k = 1, . . . , m.Fix n and denote ek = enk for brevity.Assume that 3k < n;i.e.,k < n

3.Then the line segments P0P2k and PkPn−k intersect; let Qk be their intersectionpoint and denote xk = P0Qk. Because 4QkP0Pk ∼ 4QkP2kPn−k, we have

xke2k − xk = ek

e3k.

Hence

xk = eke2kek + e3k

= 2 sinkπn sin2kπ

nsinkπ

n + sin3kπn

=

2 sinkπn sin2kπ

nsin(2kπ

n − kπn ) + sin(2kπ

n + kπn ) = sinkπ

n sin2kπn

sin2kπn coskπ

n= tankπ

n .

If n is odd, then

tankπn = cot π

2 − kπ2m + 1 = cot[2(m − k) + 1]π

2n = sn,m−k+1.


Thus sn,m−k+1= P0Qk, k = 1, . . . , bn−13 c. In other words,the bn−1

3 c smallestzeros of Vm(x) are the squared line segments P0Qk, k = 1, . . . , bn−1

3 c. Musto-nen [3] found this experimentally. The largest zero is already interpreted, butthe interpretation ofthe rest ofzeros remains open.For some experimentalobservations, see [3]. Interpretation of the zeros of Um(x), except the largest,remains open, too.

5 Expressing sn1, . . . , snm using en1, . . . , enm

Mustonen’s [3] experiments make conjecture that, given n, there are numbersλ(i)

nk ∈ 0, ±1, i, k = 1, . . . , m, such that

sni = λ(i)n1en1 + · · · + λ(i)n,m−1en,m−1+ λ(i)

nme0nm, i = 1, . . . , m,

wheree0

nm =12enm if n is even,enm if n is odd.

In other words,

cot(2i − 1)π2n = 2

hλ(i)

n1 sinπn + · · · + λ(i)n,m−1sin(m − 1)π

n + θnλ(i)nm sinmπ

ni,

whereθn =

12 if n is even,1 if n is odd.

This is true by (12) when i = 1 (sn1 = sn, λ(1)n1 = · · · = λ(1)

nm = 1) but remainsgenerally open.

For example, let n = 15. Denoting sk = s15,k and ek = e15,k for brevity, wehave [3, p. 17]

s1 = e 1+ e 2+ e3+ e4+ e 5+ e 6+ e7s2 = e3+ e6s3 = e5s4 = e 1− e 2+ e3− e4+ e 5− e 6+ e7s5 = −e3+ e6s6 = e 1− e 2+ e3+ e4− e 5+ e 6− e7s7 = e 1+ e 2− e3− e4+ e 5+ e 6− e 7.


We study the zero coefficients in general. If and only if d = gcd(n, 2i−1) > 1,then Gn ”inherits” the chord

sni = cot(2i − 1)π2n

from Gd. Then the chords of Gd are enough to express sni , and the coefficientsof the remaining chords are zero. Indeed, in our example,

s2 = s15,2= cot3π30 = cotπ

10= 2 sinπ5 + sin2π

5 ,

s3 = s15,3= cot5π30 = cotπ6 = 2 sinπ3,

s5 = s15,5= cot9π30 = cot3π

10 = 2 − sinπ5 + sin2π

5 ,showing that s3 is ”inherited” from G3, and s2 and s5 from G5.

So we conjecture additionally that ifand only ifn is a prime or a powerof 2, then each λ(i)

nk ∈ ±1. Mustonen [3] gives also other experimental resultsand conjectures about the structure of the three-dimensional array (λ(i)

nk), andpresents an efficient algorithm to compute these numbers.

6 Connections with OEIS sequencesThe (lexicographically ordered) sequence (amk) is A053122 in OEIS. Its firstsix terms are a00= 1, a10= −2, a11= 1, a20= 3, a21 = −4, a22= 1.

The OEIS sequence A132460 consists of the numberstn0 = 1, n = 0, 1, 2, . . . ,

tnk = (−1)k( n − kk + n − k − 1

k − 1 ), n = 2, 3, . . . ,k = 1, . . . , m.

The first six terms ofits subsequence corresponding to odd values ofn aret10 = 1 = b00, t30 = 1 = b11, t31 = −3 = b10, t50 = 1 = b22, t51 = −5 = b21,t52= 5 = b20. In general, bmk = t2m+1,m−k.

Also the characteristic polynomials ofcertain other tridiagonalmatriceshave connections with OEIS sequences. We study two of them.

Let tridiag(a, b, c) denote the tridiagonal matrix with main diagonal, sub-diagonaland superdiagonalentries those of vectors a,b and c,respectively,and denote x(k) = x, . . . , x, k copies. For m ≥ 3, define

Cm = tridiag ((2(m)), ((−1)(m−2), −2), (−2, (−1)(m−2)))


andC2 = 2 −2

−2 2 , C1 = (2).

For m ≥ 1, consider the polynomial

Cm(x) = det (xIm − Cm) = xm + cm,m−1xm−1+ · · · + cm1x + cm0

and define C0(x) = 1, c00 = cmm = 1. The sequence A140882 consists ofthe numbers (−1)mcmk. Since C0(x) = 1, C1(x) = x − 2,C2(x) = x2 − 4x,C3(x) = x3 − 6x2 + 8x,its first ten terms are 1, 2, −1, 0, −4, 1, 0, −8, 6, −1,aslisted in [4].

We have xA1(x) = x2 − 4x = C2(x) and xA2(x) = x3 − 6x2 + 8x = C3(x),and generally

Cm+1(x) = xAm(x) (19)

for all m ≥ 1. This can be proved similarly to the proofs of Theorems 1 and 2.By (8), a formula for A140882 is then obtained. By (19), (7) and (3),

spec Cm = spec Am−2∪ 0, 4 =4 sin2 kπ2m − 2 k = 0, . . . , m − 1

for m ≥ 3.Finally, the sequence A136672 motivates us to study the polynomial

Fm+1(x) = (x − 2)Am(x) = xm+1+ fm+1,mxm + · · · + fm+1,1x + fm+1,0 (20)

and its connections with the matrix Dm, defined by

Dm = tridiag((2(m)), ((−1)(m−2), 0), ((−1)(m−1)))

if m ≥ 3, andD2 = 2 −1

0 2 , D1 = (2).

By Theorem 1,

fm+1,k= (−1)m−k+1( m + k2k − 1 + 2 m + 1 + k

2k + 1 ). (21)

For m ≥ 1, consider the polynomial

Dm(x) = det (xIm − Dm) = xm + dm,m−1xm−1+ · · · + dm1x + dm0


and define D0(x) = 1, d00 = dmm = 1. The sequence A136672 consists of thenumbers (−1)mdmk. We have D0(x) = 1, D1(x) = x − 2, D2(x) = x2 − 4x + 4,D3(x) = x3−6x2+11x−6. So its first ten terms are 1, 2, −1, 4, −4, 1, 6, −11, 6, −1,as listed in [4].

Since F1(x) = x − 2 = D1(x), F2(x) = x2 − 4x + 4 = D2(x), and F3(x) =x3 − 6x2 + 11x − 6 = D3(x), it seems that

Dm(x) = Fm(x) (22)

for all m ≥ 1. This can be proved similarly to the previous proofs. By (21), aformula for A136672 follows. By (22), (20) and (3),

spec Dm = spec Am−1∪ 2 =4 sin2 kπ2m k = 1, . . . , m − 1∪ 2

for m ≥ 2.

7 Conclusions and further questionsThe squared chords of Gn, excluding the diameter,are the zeros of a monicpolynomialof degree µ with integer coefficients.Including the diameter,thedegree is m.

The squared sum ofall individualchords is the largest zero ofa monicpolynomialof degree m with integer coefficients.An equivalent fact is thatthe squared sum of all different (lengths of) chords but the diameter is halved,is a zero of such a polynomial. The zeros of this polynomial seem to be linearcombinations of the chords with all coefficients 0 or ±1.

Lemma 1, stating that e2n1, . . . , e2nµ are the eigenvalues of a tridiagonal ma-

trix with integer entries,follows from certain properties ofthe Chebychevpolynomials.So squared chords have interesting connections with these top-ics. But what about s2n1, . . . , s2

nm? Are also they the eigenvalues ofsuch atridiagonal matrix? This question remains open.

The coefficients of the polynomial (x − e2n1) · · · (x − e2

nµ) form an OEIS se-quence, and so do also those of certain related polynomials. What about thecoefficients of (x − s2

n1) · · · (x − s2nm)? Do also they form such a sequence? This

question remains open, too.


Appendix: Proofs of (6) and (11)Proof of (6)

(x − 2)Pm(x) − Pm−1(x)

= (x − 2)mX

k=0(−1)m−k m + 1 + k

2k + 1 xk −m−1X

k=0(−1)m−1−k m + k

2k + 1 xk

− xm+1+m−1X

k=0(−1)m−k m + 1 + k

2k + 1 xk+1− 2mX

k=0(−1)m−k m + 1 + k

2k + 1 xk

−m−1X

k=0(−1)m−1−k m + k

2k + 1 xk

= xm+1+mX

k=1(−1)m+1−k m + k

2k − 1 xk + 2mX

k=0(−1)m+1−k m + 1 + k

2k + 1 xk

−m−1X

k=0(−1)m+1−k m + k

2k + 1 xk

= xm+1− 2m2m − 1 + 2 2m + 1

2m + 1 xm

+m−1X

k=1(−1)m+1−k m + k

2k − 1 + 2 m + 1 + k2k + 1 − m + k

2k + 1 xk

+ (−1)m+1 2 m + 11 − m

1

= xm+1− (2m + 2)xm +m−1X

k=1(−1)m+1−k m + 2 + k

2k + 1 xk + (−1)m+1(m + 2)

=m+1X

k=0(−1)m+1−k m + 1 + 1 + k

2k + 1 xk = Pm+1(x).


Proof of (11)

(x − 3)Pm(x) − Pm−1(x)

= · · · = xm+1− 2m2m − 1 + 3 2m + 1

2m + 1 xm

+m−1X

k=1(−1)m+1−k m + k

2k − 1 + 3 m + 1 + k2k + 1 − m + k

2k + 1 xk

+ (−1)m+1 3 m + 11 − m

1

= xm+1− (2m + 3)xm +m−1X

k=1(−1)m+1−k 2m + 3

m − k + 1m + 1 + k

2k + 1 xk

+ (−1)m+1(2m + 3)

= xm+1+mX

k=0(−1)m+1−k2(m + 1) + 1

m + 1 − km + 1 + k

2k + 1 xk = Qm+1(x).

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Received: 4 August 2014


On the weighted integral inequalities forconvex function

Mehmet Zeki SarikayaDepartment of Mathematics,Faculty of Science and Arts,

Duzce University, Konuralp Campus,Duzce-Turkey


Samet ErdenDepartment of Mathematics,

Faculty of Science,Bartin University,

Bartin-Turkeyemail: [email protected]

Abstract. In this paper,we establish severalweighted inequalities forsome differantiable mappingsthat are connected with the celebratedHermite-Hadamard-Fej´er type and Ostrowskitype integralinequalities.The results presented here would provide extensions ofthose given inearlier works.

1 IntroductionThe following result is known in the literature as Ostrowski’s inequality [10]:

Theorem 1 Letf : [a, b]→ R be a differentiable mapping on (a, b) whosederivative f0 : (a, b)→ R is bounded on (a, b),i.e., kf0k∞ = sup

t∈(a,b)|f0(t)| < ∞.

Then, the inequality:

f(x) − 1b − a

bZ

af(t)dt ≤

"14 + (x −a+b

2 )2

(b − a)2#

(b − a) f0∞ (1)

holds for allx ∈ [a, b]. The constant14 is the best possible.

2010 Mathematics Subject Classification: 26D07, 26D15Key words and phrases: Ostrowski’s inequality, Montgomery’s identities, convex function,Holder inequality

194

DOI: 10.1515/ausm-2015-0006

On the weighted integral inequalities 195

Inequality (1) has wide applications in numericalanalysis and in the the-ory of some specialmeans;estimating error bounds for some specialmeans,some mid-point, trapezoid and Simpson rules and quadrature rules, etc. Henceinequality (1) has attracted considerable attention and interest from mathe-maticans and researchers. Due to this, over the years, the interested reader isalso refered to ([1]-[7],[12]-[17]) for integral inequalities in several independentvariables. In addition, the current approach of obtaining the bounds, for a par-ticular quadrature rule, have depended on the use of Peano kernel. The generalapproach in the past has involved the assumption of bounded derivatives ofdegree greater than one.

If f : [a, b] → R is differentiable on [a, b] with the first derivative f0integrableon [a, b], then Montgomery identity holds:

f(x) = 1b − a

bZ

af(t)dt +

bZ

aP(x, t)f0(t)dt, (2)

where P(x, t) is the Peano kernel defined by

P(x, t) :=

t − ab − a, a ≤ t < x

t − bb − a, x ≤ t ≤ b.

Definition 1 The function f :[a, b]⊂ R → R, is said to be convex iftheinequality

f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y)holds for allx, y ∈ [a, b]and λ ∈[0, 1]. We say that f is concave if(−f) isconvex.

The following inequality is wellknown in the literature as the Hermite-Hadamard integral inequality (see, [11]):

f a + b2 ≤ 1

b − aZb

af(x)dx ≤ f(a) + f(b)

2 (3)holds,where f : I ⊂ R → R is a convex function on the intervalI of realnumbers and a, b ∈ I with a < b.

The most well-known inequalities related to the integralmean of a convexfunction are the Hermite-Hadamard inequalities or its weighted versions, theso-called Hermite-Hadamard-Fej´er inequalities (see, [18]-[22]). In [8], Fej´er gavea weighted generalization of the inequality (3) as the following:

196 M. Z. Sarikaya, S. Erden

Theorem 2 Let f : [a, b] → R, be a convex function, then the inequality

f a + b2

Zb

aw(x)dx ≤ 1

b − aZb

af(x)w(x)dx ≤f(a) + f(b)

2Zb

aw(x)dx (4)

holds, where w : [a, b] → R is nonnegative, integrable, and symmetric regardingx = a+b

2 .In [18], some inequalities of Hermite-Hadamard-Fej´er type for differentiable

convex mappings were proved using the following lemma.Lemma 1 Let f :I ⊂ R → R be a differentiable mapping on I, a, b ∈ Iwith a < b, and w : [a, b] → [0, ∞) be a differentiable mapping. If f0∈ L[a, b],then the following equality holds:

1b − a

Zb

af(x)w(x)dx − 1

b − afa + b

2Zb

aw(x)dx

= (b − a)Z1

0k(t)f0(ta + (1 − t)b)dt

(5)

for each t ∈ [0, 1], where

k(t) =

Rt0w(as + (1 − s)b)ds, t ∈ [0,12)

−R1t w(as + (1 − s)b)ds,t ∈ [12, 1].

The main result in [18] is as follows:Theorem 3 Let f :I ⊂ R → R be a differentiable mapping on I, a, b ∈ Iwith a < b, and w : [a, b] → [0, ∞) be a differentiable mapping and symmetricto a+b

2 . If |f0| is convex on[a, b], then the following inequality holds:

1b − a

Zb

af(x)w(x)dx − 1

b − afa + b

2Zb

aw(x)dx

(6)

≤

1(b − a)2

Zb

a+b2

w(x)h(x − a)2 −(b − x)2

idx

!|f0(a)|+ |f0(b)|

2 .

In this article, using functions whose derivatives absolute values are convex,we obtained new inequalities of Fejer-Hermite-Hadamard type and Ostrowskitype.The results presented here would provide extensions ofthose given inearlier works.


2 Main resultsWe will establish some new results connected with the left-hand side of(4)and Ostrowski type inequalities used the following Lemma. Now, we give thefollowing new Lemma for our results:Lemma 2 Let f :I ⊆ R → R be a differentiable mapping on I, a, b ∈ Iwith a < b and let w :[a, b] → R. If f0, w ∈ L[a, b],then,for all x ∈ [a, b],the following equality holds:

xZ

a

tZ

aw(s)ds

α

f0(t)dt −bZ

x

bZ

tw(s)ds

α

f0(t)dt

=

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x)

− αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt − αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt.

(7)

Proof. By integration by parts, we have the following equalities:xZ

a

tZ

aw(s)ds

α

f0(t)dt =

tZ

aw(s)ds

α

f(t)x

a

− αxZ

a

tZ

aw(s)ds

w(t)f(t)dt

=

xZ

aw(s)ds

α

f(x) − αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt

(8)

andbZ

x

bZ

tw(s)ds

α

f0(t)dt

=

bZ

tw(s)ds

α

f(t)b

x

+ αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt

= −

bZ

xw(s)ds

α

f(x) + αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt.

(9)


Subtracting (8) from (9), we obtain (7)

xZ

a

tZ

aw(s)ds

α

f0(t)dt −bZ

x

bZ

tw(s)ds

α

f0(t)dt

=

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x)

−αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt − αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt.

This completes the proof.

Corollary 1 Under the same assumptions as in Lemma 2, if we put α = 1,then the following identity holds:

bZ

aw(s)ds

f(x) −bZ

aw(t)f(t)dt

=xZ

a

tZ

aw(s)ds

f0(t)dt −bZ

x

bZ

tw(s)ds

f0(t)dt(10)

Remark 1 If we take w(s)= 1 in (10), the idendity (10) reduces to theidentity (2).

Definition 2 Let f ∈ L1[a, b]. The Riemann-Liouville integrals Jαa+f and Jαb−f

of order α > 0 with a ≥ 0 are defined by

Jαa+ f(x) = 1

Γ (α)Zx

a(x − t)α−1f(t)dt, x > a

and

Jαb− f(x) = 1

Γ (α)Zb

x(t − x)α−1f(t)dt, x < b

respectively. Here, Γ (α) is the Gamma function and J0a+ f(x) = J0b− f(x) = f(x).


Corollary 2 Under the same assumptions as in Lemma 2, if we put w(s) = 1,then the following equality holds:

[(x − a)α + (b − x)α]f(x) − Γ (α + 1)Jαx− f(a) − Γ (α + 1)Jα

x+ f(b) (11)

=xZ

a(t − a)α f0(t)dt −

bZ

x(b − t)α f0(t)dt.

Corollary 3 Under the same assumptions ofCorollary 2 with x =a+b2 , the

idendity (11) becomes to the following identity

f a + b2 − Γ (α + 1)

21−α(b − a)α Jα( a+b

2 )− f(a) + Jα( a+b2 )+ f(b)

= 121−α(b − a)α

a+b2Z

a(t − a)α f0(t)dt −

bZ

a+b2

(b − t)α f0(t)dt

.

Now, by using the above lemma, we prove our main theorems:

Theorem 4 Let f :I ⊆ R → R be a differentiable mapping on I, a, b ∈ Iwith a < b and let w :[a, b] → R be continuous on[a, b]. If |f0| is convex on[a, b], then the following inequality holds:

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x)

−αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt − αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt

≤kwkα

[a,x],∞b − a

(b − a)(x − a)α+1

α + 1 − (x − a)α+2

α + 2 |f0(a)|

+kwkα

[a,x],∞b − a

(x − a)α+2

α + 2 |f0(b)| +kwkα

[x,b],∞b − a

(b − x)α+2

α + 2 |f0(a)|

+kwkα

[a,x],∞b − a

(b − a)(b − x)α+1

α + 1 − (b − x)α+2

α + 2 |f0(b)|


≤kwkα

[a,b],∞b − a

(b − a)(x − a)α+1

α + 1 + (b − x)α+2− (x − a)α+2

α + 2 |f0(a)|

+ (b − a)(b − x)α+1

α + 1 + (x − a)α+2− (b − x)α+2

α + 2 |f0(b)|

where α > 0 and kwk[a,b],∞ = supt∈[a,b]

|w(t)|.

Proof. We take absolute value of both sizes of (7), we find that

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x)

−αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt − αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt

≤xZ

a

tZ

aw(s)ds

α

f0(t) dt +bZ

x

bZ

tw(s)ds

α

|f0(t)|dt

≤ kwkα[a,x],∞

xZ

a(t − a)α f0(t) dt + kwk[x,b],∞

bZ

x(b − t)α |f0(t)|dt

= kwkα[a,x],∞

xZ

a(t − a)α f0( b − t

b − aa + t − ab − ab) dt

+ kwkα[x,b],∞

bZ

x(b − t)α f0( b − t

b − aa + t − ab − ab) dt

Since |f0| is convex on [a.b], it follows that

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x)

−αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt − αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt


≤ kwkα[a,x],∞

xZ

a(t − a)α b − t

b − a f0(a) + t − ab − a|f0(b)| dt

+ kwkα[x,b],∞

bZ

x(b − t)α b − t

b − a|f0(a)| +t − a

b − a|f0(b)| dt

=kwkα

[a,x],∞b − a

(b − a)(x − a)α+1

α + 1 − (x − a)α+2

α + 2 |f0(a)| +(x − a)α+2

α + 2 |f0(b)|

+kwkα

[x,b],∞b − a

(b − x)α+2

α + 2 f0(a) + (b − a)(b − x)α+1

α + 1 − (b − x)α+2

α + 2 |f0(b)|

≤kwkα

[a,b],∞b − a

(b − a)(x − a)α+1

α + 1 + (b − x)α+2− (x − a)α+2

α + 2 |f0(a)|

+ (b − a)(b − x)α+1

α + 1 + (x − a)α+2− (b − x)α+2

α + 2 |f0(b)| .

Hence, the proof of theorem is completed.

Corollary 4 Under the same assumptions as in Theorem 4, if we take w(s) =1, then the following inequality holds:

|[(x − a)α + (b − x)α] f(x) − Γ (α + 1) [Jαx− f(a) + Jαx+ f(b)]|

≤ 1b − a

(b − a)(x − a)α+1

α + 1 + (b − x)α+2− (x − a)α+2

α + 2 |f0(a)|

+ (b − a)(b − x)α+1

α + 1 + (x − a)α+2− (b − x)α+2

α + 2 |f0(b)| .

(12)

Remark 2 If we take x =a+b2 in (12), we get

f a + b2 − 2α−1Γ (α + 1)

(b − a)α Jα( a+b

2 )− f(a) + Jα( a+b2 )+ f(b)

≤ (b − a)4(α + 1) f0(a) + f0(b)

which is proved by Sarikaya and Yildirim in [19].


Corollary 5 Under the same assumptions as in Theorem 4, if we take α = 1,then the following inequality holds:

bZ

aw(s)ds

f(x) −bZ

aw(t)f(t)dt

≤kwk[a,b],∞

b − a(b − a)(x − a)2

2 + (b − x)3 − (x − a)33 |f0(a)|

+ (b − a)(b − x)22 + (x − a)3 − (b − x)3

3 |f0(b)| .

Corollary 6 Under the same assumptions ofCorollary 5 with x =a+b2 , we

get

bZ

aw(s)ds

f a + b2 −

bZ

aw(t)f(t)dt

≤(b − a)2kwk[a,b],∞

4

f0(a) + f0(b)

2

.(13)

Remark 3 If we take w(s) = 1 in (13), we have

f a + b2 − 1

b − a

bZ

af(t)dt ≤ (b − a)

4

f0(a) + f0(b)

2

which is proved by Kırmacı in [9].

Corollary 7 Under the same assumptions as in Theorem 4, if we put |f0(a)|=|f0(b)| in (10), then the following inequality holds:

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x)

−αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt − αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt


≤f0(a) kwkα

[a,x],∞α + 1 (x − a)α+1+

f0(a) kwkα[x,b],∞

α + 1 (b − x)α+1

≤f0(a) kwkα

[a,b],∞α + 1

h(x − a)α+1+ (b − x)α+1

i

Theorem 5 Let f :I ⊆ R → R be a differentiable mapping on I, a, b ∈ Iwith a < b and let w :[a, b] → R be continuous on[a, b]. If |f0|q is convex on[a, b], q > 1, then the following inequality holds:

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x) − αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt

−αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt ≤kwkα

[a,x],∞

(b − a)1q(x − a)αp+1

αp + 1

1p

(b − a)2 − (b − x)22 |f0(a)|q + (x − a)2

2 |f0(b)|q1q

+kwkα

[x,b],∞

(b − a)1q(b − x)αp+1

αp + 1

1p (b − x)2

2 |f0(a)|q + (b − a)2 − (x − a)22 |f0(b)|q

1q

≤kwkα

[a,b],∞

(b − a)1q

(x − a)αp+1

αp + 1

! 1p (b − a)2 − (b − x)2

2 |f0(a)|q

+(x − a)22 |f0(b)kq

1q

+

(b − x)αp+1

αp + 1

! 1p

(b − x)2

2 |f0(a)|q + (b − a)2 −(x − a)2

2 |f0(b)|q! 1

q

(14)

where α > 0,1p + 1q = 1, and kwk[a,b],∞ = sup

t∈[a,b]|w(t)|.


Proof. We take absolute value of (7). Using Holder’s inequality, we find that

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x)

−αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt − αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt

≤xZ

a

tZ

aw(s)ds

α

|f0(t)|dt +bZ

x

bZ

tw(s)ds

α

|f0(t)|dt

≤xZ

a

tZ

aw(s)ds

αp

dt

1p

xZ

a|f0(t)|qdt

1q

+bZ

x

bZ

tw(s)ds

αp

dt

1p

bZ

x|f0(t)|qdt

1q

≤ kwkα[a,x],∞

xZ

a|t − a|αp dt

1p

xZ

a|f0(t)|qdt

1q

+ kwkα[x,b],∞

bZ

x|b − t|αp dt

1p

bZ

x|f0(t)|qdt

1q

Since f0(t)q

is convex on[a, b]

f0( b − tb − aa + t − a

b − ab)q

≤ b − tb − a f0(a)

q+ t − a

b − a f0(b)q

(15)

From (15), it follows that

xZ

aw(s)ds

α

+

bZ

xw(s)ds

α

f(x)

−αxZ

a

tZ

aw(s)ds

α−1

w(t)f(t)dt − αbZ

x

bZ

tw(s)ds

α−1

w(t)f(t)dt


≤kwkα

[a,x],∞

(b − a)1q(x − a)αp+1

αp + 1

1p

(b − a)2 − (b − x)2

2 |f0(a)|q

+(x − a)22 f0(b)

q1q

+kwkα

[x,b],∞

(b − a)1q(b − x)αp+1

αp + 1

1p

(b − x)2

2 |f0(a)|q + (b − a)2 − (x − a)22 |f0(b)|q

! 1q

≤kwkα

[a,b],∞

(b − a) 1q

(x − a)αp+1

αp + 1

! 1p

(b − a)2 −(b − x)2

2 |f0(a)|q + (x − a)22 |f0(b)|q

! 1q

+

(b − x)αp+1

αp + 1

! 1p

(b − x)2

2 |f0(a)|q + (b − a)2 −(x − a)2

2 |f0(b)|q! 1

q

which completes the proof.

Corollary 8 Under the same assumptions as in Theorem 4, if we put w(s) =1, then the following inequality holds:

|[(x − a)α + (b − x)α]f(x) − Γ (α + 1)[Jαx− f(a) + Jαx+f(b)]|≤ 1

(b − a) 1q

(x − a)αp+1

αp + 1

! 1p

(b − a)2 −(b − x)2

2 |f0(a)|q + (x − a)22 |f0(b)|q

! 1q

+

(b − x)αp+1

αp + 1

! 1p

(b − x)2

2 |f0(a)|q + (b − a)2 −(x − a)2

2 |f0(b)|q! 1

q

.

(16)

Remark 4 If we take x =a+b2 in (16), we have

f a + b2 − 2α−1Γ (α + 1)

(b − a)α Jα( a+b

2 )− f(a) + Jα( a+b2 )+ f(b)

≤ (b − a)4(αp + 1)1p

3|f0(a)|q + |f0(b)|q4

! 1q

+

|f0(a)|q + 3|f0(b)|q4

! 1q

which is proved by Sarikaya and Yildirim in [19].


Corollary 9 Let the conditions of Theorem 5 hold. If we take α = 1 in (14),then the following inequality holds:

bZ

aw(s)ds

f(x) −bZ

aw(t)f(t)dt ≤

kwk[a,b],∞

(b − a) 1q

(x − a)p+1

p + 1

! 1p

(b − a)2 −(b − x)2

2 |f0(a)|q + (x − a)22 |f0(b)|q

! 1q

+

(b − x)p+1

p + 1

! 1p

(b − x)2

2 f0(a)q

+ (b − a)2 −(x − a)2

2 |f0(b)|q! 1

q

Corollary 10 Under the same assumptions of Corollary 9 with x =a+b2 , we

get

bZ

aw(s)ds

f a + b2 −

bZ

aw(t)f(t)dt ≤

(b − a)2kwk[a,b],∞

22+1q (p + 1)1p

3|f0(a)|q + |f0(b)|q

2

! 1q

+

|f0(a)|q + 3|f0(b)|q2

! 1q

.

(17)

Remark 5 If we take w(s) = 1 in (17), we have

f a + b2 − 1

b − a

bZ

af(t)dt

≤ (b − a)22+1

q (p + 1)1p

3|f0(a)|q + |f0(b)|q

2

! 1q

+

|f0(a)|q + 3|f0(b)|q2

! 1q

which is proved by Kırmacı in [9].


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Received: 16 May 2014


Evolution of =-functional and ω-entropyfunctional for the conformal Ricci flow

Nirabhra BasuDepartment of Mathematics,The Bhawanipur Education

Society College,Kolkata-700020, West Bengal, Indiaemail: [email protected]

Arindam BhattacharyyaDepartment of Mathematics

Jadavpur University,Kolkata-700032, India


Abstract. In this paper we define the =-functional and the ω-entropyfunctional for the conformal Ricci flow and see how they evolve accordingto time.

1 IntroductionIn 1982 R.Hamilton introduced Ricciflow as a deformation ofRiemannianmetric [3], [4]. After him many scientists gave attention on it and in 2003–2004G. Perelman [1],[2] used it to prove Poincar´e conjecture.Meanwhile in 2004A. E. Fischer introduced the concept of conformal Ricci flow equation whichis given by

∂g∂t + 2 S +g

n = −pgR(g) = −1.

(1)

Here p is a scalar non dynamicalfield.As conformalRicci flow equation isanalogous to the Navier-Stokes equation of fluid mechanics, the scalar field pis also called conformal pressure field.

2010 Mathematics Subject Classification: 53C44, 35K65, 58D17Key words and phrases: Ricci flow, conformal Ricci flow, entropy functional

209

DOI: 10.1515/ausm-2015-0007

210 N. Basu, A. Bhattacharyya

The name conformalRicci flow was introduced because ofthe role thatconformalgeometry plays in constraining the scalar curvature and becausethese equations are the vector field sum ofa conformalflow equation anda Ricci flow equation.For the classicalRicci flow equation and the confor-malRicci flow equation,the volume and scalar curvature behave somewhatoppositely.In classicalRicci flow equation,the volume is preserved,that isvol(M, g) = 1, but for non-static flows the scalar is not preserved, whereas forconformal Ricci flow equation the scalar curvature R(g) is kept constant to −1and for non-static flows the volume varies.Comparing the classicaland con-formal Ricci flow equations, we observe that the constraint equation changesfrom vol(M, g) = 1 for the classical Ricci flow to R(g) = −1 for the conformalRicci flow with the concomitant change of the configuration space from M1 toM −1. Since M1 is a codimension-1 submanifold of M whereas M−1 is a codi-mension C∞ (M, <) submanifold of M,M −1 is a much smaller configurationspace than M1. In the view point of geometry having a smaller configurationspace is potentially better.

From the lecture note of P. Topping [5], we have been introduced the conceptof the =-functional and Perelman’s ω entropy functional for Ricci flow. In ourpaper we have defined the =-functionaland ω-entropy functionalregardingconformal Ricci flow and have shown how they evolve with respect to time t.

2 The =-functional for the conformal Ricci flowLet M be a fixed closed manifold, g is a Riemannian metric and f is a functiondefined on M to the set of real numbers <.

Then the =-functional on pair (g, f) is defined as

=(g, f) =Z

−1 + |∇f|2 e−fdV. (2)

Now we establish how the =-functional changes according to time under con-formal Ricci flow.

Theorem 1 In conformalRicci flow, the rate of change of =-functionalwithrespect of time is given by

ddt=(g, f) =

Z−2Ric(∇f, ∇f) − 2

n + p g(∇f, ∇f) − 2∂f∂t (∆f − |∇f|2)

+ (−1 + |∇f|2) −∂f∂t + 1

2tr ∂g∂t e−fdV,

Evolution of = and ω-entropy functional for the conformal Ricci flow211

where =(g, f) =R(−1 + |∇f|2)e−fdV.

Proof.∂∂t |∇f|2 = ∂

∂t g(∇f, ∇f) = ∂g∂t (∇f, ∇f) + 2g ∇ ∂f

∂t , ∇f . (3)

So using proposition 2.3.12 of [5] we can writeddt=(g, f) =

Z ∂g∂t (∇f, ∇f) + 2g ∇ ∂f

∂t , ∇f e−fdV

+Z(−1 + |∇f|2) −∂f

∂t + 12tr ∂g

∂t e−fdV.(4)

Using integration by parts of equation (3), we getZ

2g ∇ ∂f∂t , ∇f e−fdV = −2

Z∂f∂t (∆f − |∇f|2)e−fdV. (5)

Now putting (5) in (4), we getddt=(g, f) =

Z ∂g∂t (∇f, ∇f) − ∂f

∂t (∆f − |∇f|2)

+(−1 + |∇f|2) −∂f∂t + 1

2tr ∂g∂t e−fdV.

(6)

Using (1) in (6), we get the following result for conformal Ricci flow, asddt=(g, f) =

Z− 2Ric(∇f, ∇f) − 2

n + p g(∇f, ∇f)

− 2∂f∂t (∆f − |∇f|2) + (−1 + |∇f|2) −∂f

∂t + 12tr ∂g

∂t

e−fdV.(7)

Hence the proof.

3 ω-entropy functional for the conformal Ricci flowLet M be a closed manifold, g is a Riemannian metric on M and f is a smoothfunction defined from M to the set of realnumbers <.We define ω-entropyfunctional as

ω(g, f, τ) =Zh

τ −1 + |∇f|2 + f − ni

udV, (8)


where τ > 0 is a scale parameter and u is defined as u(t) = e−f(t);RM udV = 1.We would also like to define heat operator acting on the function f :M ×

[0, τ] −→ < by ♦ :=∂∂t − ∆ and also, ♦∗:= −∂∂t − ∆ − 1, conjugate to ♦.

We choose u, such that ♦∗u = 0.Now we prove the following theorem.

Theorem 2 If g, f, τ evolve according to

∂g∂t = −2Ric − 2

n + p g (9)∂τ∂t = −1 (10)∂f∂t = −∆f + |∇f|2 + 1 +n

2τ (11)

and the function v defined as v = [τ(2∆f − |∇f|2− 1) + f − n]u, then the rate ofchange of ω-entropy functionalfor conformalRicci flow is dω

dt = −RM ♦∗v,

where

♦∗v = 2u(∆f − |∇f|2 − 1) −un2τ − v − uτ[4 < Ric, Hessf >

+ 2n + p g(∇f, ∇f) − 2g(∇|∇f|2, ∇f) + 4g(∇(∆f), ∇f) + 2|Hessf|2].

Proof.♦∗v = ♦∗ v

uu = vu♦∗u + u♦∗ v

u .We have defined previously that ♦∗u = 0, so

♦∗v = u♦∗ vu

♦∗v = u♦∗[τ(2∇f − |∇f|2 − 1) + f − n].

We shall use the conjugate ofheat operator,as defined earlieras ♦∗ =− ∂

∂t + ∆ + 1. Therefore ♦∗v = −u ∂∂t + ∆ + 1[τ(2∆f−|∇f|2−1)+f−n] ⇒

u−1♦∗v = − ∂∂t + ∆ [τ(2∆f−|∇f|2−1)]− ∂

∂t + ∆ f−[τ(2∆f−|∇f|2−1)+f−n]using equation (10), we have

u−1♦∗v = (2∆f − |∇f|2 − 1) − τ ∂∂t + ∆ (2∆f − |∇f|2 − 1)

− ∂f∂t − ∆f − v

u.(12)


Now using the equality∂∂t (2∆f − |∇f|2− 1) = 2∂∂t (∆f) − ∂

∂t |∇f|2 and the propo-sition 2.5.6 of [5], we have

∂∂t (2∆f−|∇f|2−1) = 2∆∂f

∂t +4 < Ric, Hessf > −∂g∂t (∇f, ∇f)−2g ∂

∂t ∇f, ∇f .

Now using the conformal Ricci flow equation (1), we have

∂∂t (2∆f − |∇f|2 − 1) = 2∆∂f

∂t + 4 < Ric, Hessf > +2Ric(∇f, ∇f)

+ 2n + p g(∇f, ∇f) − 2g ∂

∂t ∇f, ∇f .(13)

Using (11) in (13), we get

∂∂t (2∆f − |∇f|2 − 1) = 2∆ −∆f + |∇f|2 + 1 +n

2τ+ 4 < Ric, Hessf > +2Ric(∇f, ∇f)

+ 2n + p g(∇f, ∇f) − 2g ∂

∂t ∇f, ∇f .(14)

Now let us compute

∆(2∆f − |∇f|2 − 1) = 2∆2f − ∆|∇f|2. (15)

Using (14) and (15) in (12) we obtain after a brief calculation

u−1♦∗v = (2∆f − |∇f|2 − 1) − τ− 2∆2f + 2∆|∇f|2

+ 4 < Ric, Hessf > +2Ric(∇f, ∇f)

+ 2n + p g(∇f, ∇f) − 2g ∂

∂t ∇f, ∇f

+ 2∆2f − ∆|∇f|2] −∂f∂t − ∆f − v

u= ∆f − |∇f|2 − 1 − τ[∆|∇f|2 + 4 < Ric, Hessf > +2Ric(∇f, ∇f)

+ 2n + p g(∇f, ∇f) − 2g ∂

∂t ∇f, ∇f − ∂f∂t − v

u


= ∆f − |∇f|2 − 1 − τ∆|∇f|2 + 4 < Ric, Hessf > +2Ric(∇f, ∇f)

+ 2n + p g(∇f, ∇f) − 2g ∂

∂t ∇f, ∇f

+ ∆f − |∇f|2 − 1 −n2τ− v

u= 2(∆f − |∇f|2 − 1) −n

2τ− vu − τ ∆|∇f|2 + 4 < Ric, Hessf > +2Ric(∇f, ∇f)

+ 2n + p g(∇f, ∇f) − 2g ∂

∂t ∇f, ∇f

u−1♦∗v = 2(∆f − |∇f|2 − 1) −n2τ− τ(2∆f − |∇f|2 − 1) + f − n] − τ[∆|∇f|2


+ 2n + p g(∇f, ∇f) − 2g ∂

∂t ∇f, ∇f

u−1♦∗v = 2(∆f − |∇f|2 − 1) −n2τ− f + n − τ2∆f − |∇f|2 − 1 + ∆|∇f|2


+ 2n + p g(∇f, ∇f) − 2g ∇ ∂f

∂t , ∇f (16)

using (11), we get

u−1♦∗v = 2 ∆f − |∇f|2 − 1 − n2τ− f + n − τ2∆f − |∇f|2 − 1 + ∆|∇f|2

+ 4 < Ric, Hessf > +2Ric(∇f, ∇f) +2n + p g(∇f, ∇f)

− 2g ∇ −∆f + |∇f|2 + n2τ+ 1 , ∇f .

(17)

We can rewrite (17) in the following way

u−1♦∗v = 2(∆f − |∇f|2 − 1) −n2τ− f + n − τ[2∆f − |∇f|2 − 1

+ 4 < Ric, Hessf > +2n + p g(∇f, ∇f)

− 2g(∇|∇f|2, ∇f) + 4g(∇(∆f), ∇f)]+ τ[−∆|∇f|2 − 2Ric(∇f, ∇f) + 2g(∇(∆f), ∇f)]

(18)


References[1]G. Perelman,The entropy formula for the Ricciflow and its geometric

applications, arXiv.org/abs/math/0211159, (2002) 1–39.[2]G. Perelman,Ricci flow with surgery on three manifolds,arXiv.org/

abs/math/0303109, (2002), 1–22.[3]R. S. Hamilton, Three Manifold with positive Ricci curvature, J. Differ-

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Society Science Press, 2006.[5]P. Topping, Lecture on The Ricci Flow, Cambridge University Press, 2006.[6]A. E. Fischer, An introduction to conformal Ricci flow, Class. Quantum

Grav., 21 (2004), S171–S218

Received: 13 May, 2013

ContentsVolume 6, 2014

S. D. Purohit, R. K. RainaSome classes of analytic and multivalent functions associated withq-derivative operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

G. C. RanaHydromagnetic thermoslutal instability of Rivlin-Ericksen rotatingfluid permeated with suspended particles and variable gravity fieldin porous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

T. M. Seoudy, M. K. AoufSome applications ofdifferentialsubordination to certain subclassof p-valent meromorphic functions involving convolution . . . . . . . 46

Cs. SzántóOn some Ringel-Hall numbers in tame cases . . . . . . . . . . . . . . . . . . . . 61

A. SzázA particular Galois connection between relations and setfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

I. Szöll˝osiOn the combinatoricsof extensionsof preinjectiveKroneckermodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

E. WolfComposition followed by differentiation between weighted Bergmanspaces and weighted Banach spaces of holomorphic functions . . 107

217

R. S. Batahan, A. A. BathanyaOn generalized Laguerre matrix polynomials . . . . . . . . . . . . . . . . . . 121

B. A. Bhayo, L. YinLogarithmic mean inequality for generalized trigonometric andhyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

A. A. Bouchentouf, H. SakhiStabilizing priority fluid queueing network model. . . . . . . . . . . . . 146

S. S. DragomirSome inequalities of Furuta’s type for functions of operators definedby power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

S. Mustonen, P. Haukkanen, J. MerikoskiSome polynomials associated with regular polygons . . . . . . . . . . . . 178

M. Z. Sarikaya, S. ErdenOn the weighted integral inequalities for convex function . . . . . . 194

N. Basu, A. BhattacharyyaEvolution of =-functional and ω-entropy functional for the conformalRicci flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Contents of volume 6, 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

218

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