beta-type functions and the harmonic mean · bivariable mean in (a,∞). our main result says that...

Aequat. Math. 91 (2017), 1041–1053c© The Author(s) 2017. This article is anopen access publication0001-9054/17/061041-13published online October 23, 2017https://doi.org/10.1007/s00010-017-0498-3 Aequationes Mathematicae

Beta-type functions and the harmonic mean

Martin Himmel and Janusz Matkowski

Abstract. For arbitrary f : (a,∞) → (0,∞) , a ≥ 0, the bivariable function Bf : (a,∞)2 →(0,∞) , related to the Euler Beta function, is considered. It is proved that Bf is a mean iff itis the harmonic mean H. Some applications to the theory of iterative functional equationsare given.

Mathematics Subject Classification. Primary: 33B15, 26B25, 39B22.

Keywords. Beta function, Beta-type function, Mean, Harmonic mean, Convex function,

Wright convex function, Functional equation.

1. Introduction

For a > 0 and f : (a,∞) → (0,∞) , the two variable function Bf : (a,∞)2 →(0,∞) defined by

Bf (x, y) =f (x) f (y)f (x + y)

, x, y > a,

is called a beta-type function, and f is called its generator ([1]). The notionof beta-type functions arises from the well-known relation between the Betafunction B : (0,∞)2 → (0,∞) and the Euler Gamma function Γ : (0,∞) →(0,∞) ,

B (x, y) =Γ (x) Γ (y)Γ (x + y)

, x, y > 0,

which can be written in the form BΓ = B.In this paper we are interested in answering when a beta-type function is a

bivariable mean in (a,∞). Our main result says that the beta-type function ofa generator f : (a,∞) → (0,∞) is a mean iff f (x) = 2xeα(x) where α : R → R

is an additive function, or equivalently, that Bf is the harmonic mean (Theo-rem 2). This substantially improves the result of [1] where the homogeneity ofthe beta-type function is assumed.

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1042 M. Himmel, J. Matkowski AEM

In the preliminary Sect. 2 we recall the notions of mean, premean, reflexivityof a function, and some of their properties. The increasingness of the beta-typefunction Bf is equivalent to the concavity of the function log ◦f in the senseof Wright (Proposition 1). In Sect. 3 we determine the class of all generators ffor which Bf is reflexive, that is Bf (x, x) = x, or equivalently, f satisfies theiterative functional equation [f (x)]2 = xf (2x) (Theorem 1). Applying this, inSect. 4 , we prove Theorem 2, our main result. This allows us to conclude thatevery logarithmically convex function satisfying the above iterative functionalequation must be of the form f (x) = 2xcx for some c > 0 (Theorem 3). Thefunctional equation f (2x) = x [f (x)]2 (related to beta-type functions) is alsoconsidered. We note that the Krull method [2,3] allows us to figure out aunique solution of this iterative functional equation in a more specific class offunctions (cf. Remark 7). At the end we propose a unique and natural extensionof the harmonic bivariable mean to R

2.

The case of k-variable beta-type functions, k ≥ 3, will be considered in ournext paper.

2. Preliminaries

We start with some definitions.

Definition 1. Let I ⊂ R be an interval. A function M : I2 → R is called amean in an interval I, if

min (x, y) ≤ M (x, y) ≤ max (x, y) , x, y ∈ I.

Remark 1. Let I ⊂ R be an interval and let M : I2 → R. The followingconditions are equivalent

(i) M is a mean in I;(ii) M

(J2

) ⊂ J for every subinterval J ⊂ I.

Remark 2. If M is a mean in an interval I, then it is reflexive, i.e.

M (x, x) = x, x ∈ I.

Clearly, if M is a mean in I, then M : I2 → I.

Definition 2. Let I ⊂ R be an interval. A function M : I2 → I is called apremean if it is reflexive.

Let us note the following obvious

Remark 3. Let I ⊂ R be an interval. If a function M : I2 → R is reflexive andincreasing with respect to each of the variables, then M is a mean in I.

Vol. 91 (2017) Beta-type functions and the harmonic mean 1043

Definition 3. Let I = (a,∞) if a ≥ 0, or I = [a,∞) if a > 0. For a functionf : I → (0,∞) we call Bf : I2 → (0,∞) defined by

Bf (x, y) =f (x) f (y)f (x + y)

, x, y ∈ I,

the beta-type function of generator f .

Note that the beta-type function Bf of a generator f : (a,∞) → (0,∞) isreflexive iff [f (x)]2 = xf (2x) for all x > a.

In this context one could also consider the functions of beta-type of thegenerators f defined on the intervals (−∞, a) with values in (−∞, 0) .

Remark 4. Replacing f in this Definition 3 by 1f we get

B 1f

(x, y) =f (x + y)f (x) f (y)

, x, y ∈ I,

and B 1f

is reflexive iff f (2x) = x [f (x)]2 for all x ∈ I.

We shall prove the following

Proposition 1. Let f : (0,∞) → (0,∞) be a continuous function. The followingtwo conditions are equivalent:(i) the beta-type function Bf : (0,∞)2 → (0,∞) is an increasing mean;(ii) the function log ◦f is concave (that is f is logarithmically concave) and

[f (x)]2 = xf (2x) , x > 0.

Proof. Assume (i). Then, as log is increasing in (0,∞) , the function log ◦Bf isincreasing in each variable. So, by the definition of Bf , for all x, y, z ∈ (0,∞),

x<z =⇒ log f (x)+log f (y)−log f (x + y) ≤ log f (z)+log f (y)−log f (z + y) ,

or, equivalently, for all x, y, z ∈ (0,∞),

x < z =⇒ log f (x) + log f (z + y) ≤ log f (z) + log f (x + y) .

Choosing arbitrary u, v > 0, u < v, and t ∈ (0, 1), and taking

x := u, y := (1 − t) (v − u) , z := v − (1 − t) (v − u) ,

we obtain that the above implication is equivalent to the following one: for allu, v > 0, and t ∈ (0, 1),

u < v =⇒ log f (u) + log f (v) ≤ log f (tu + (1 − t) v) + log f ((1 − t) u + tv) ,

which shows that Bf is increasing if, and only if, log ◦f is concave in the senseof Wright. Since f is continuous, in view of a theorem of Ng [6], log◦f is Wrightconcave if, and only if, log ◦f is concave. Since Bf is a mean, it is reflexive.Thus we have shown that (ii) holds true. The converse implication follows fromRemark 3. �


3. Beta-type functions and reflexivity

Applying the method of the theory of iterative functional equations by Kuczma[4], we prove the following

Theorem 1. Assume that a > 0 is fixed.(i) Let f : [a,∞) → (0,∞) be arbitrary and let f0 := f |[a,2a) . The function

f satisfies the functional equation

[f (x)]2 = xf (2x) , x ≥ a, (3.1)

if, and only if,

f (x) = 21+(n−1)2n

x1−2n(f0

( x

2n

))2n

, x ∈ [2na, 2n+1a

), n ∈ N0. (3.2)

(ii) Let f : (0,∞) → (0,∞) be arbitrary and let f0 := f |[a,2a) . Thefunction f satisfies the functional equation

[f (x)]2 = xf (2x) , x > 0, (3.3)

if, and only if,

f (x) = 21+(n−1)2n

x1−2n(f0

( x

2n

))2n

, x ∈ [2na, 2n+1a

), n ∈ Z. (3.4)

(iii) Moreover, in each of the above cases, f is continuous if, and only if, f0

is continuous and

f (2a−) := limx→2a−

f0 (x) = (f0 (a))2 . (3.5)

Proof. Without any loss of generality we can assume that a = 1. Indeed, afunction f : [a,∞) → (0,∞) satisfies

[f (x)]2 = xf (2x) , x ≥ a,

if, and only if, the function g : [1,∞) → (0,∞) defined by

g (x) =f (ax)

a, x ≥ 1,

satisfies the equation

[g (x)]2 = xg (2x) , x ≥ 1.

(i) Let f : [1,∞) → (0,∞) be a solution to (3.1) and put f0 := f |[1,2) . Weshall show that, for every n ∈ N0,

x ∈ [2n, 2n+1

) ⇒ f (x) = 21+2n(n−1)x1−2n(f0

( x

2n

))2n

. (3.6)

Take x ≥ 1. Then there exists a unique n ∈ N0 such that

2n ≤ x < 2n+1.


If n = 0, then 1 ≤ x < 2 and, by the definition of f0,

f (x) = f0 (x) ,

so (3.6) holds true for n = 0. Assume that (3.6) holds true for x ∈ [2n, 2n+1

)

and take arbitrary x ∈ [2n+1, 2n+2

). Then x

2 ∈ [2n, 2n+1

)and, by (3.6),

f(x

2

)= 2n2n

x1−2n(f0

( x

2n+1

))2n

. (3.7)

Replacing x by x2 in (3.1), we obtain

f(x

2

)=

(x

2f (x)

) 12

,

and thus

f (x) =2x

(f

(x

2

))2

.

Using this and our assumption that f satisfies (3.1), we have

f (x) =2x

(f

(x

2

))2

=2x

(2n2n

x1−2n(f0

( x

2n+1

))2n)2

= 21−2−(n+1)x1−2−(n+1)

(f0

( x

2n+1

))2n+1

,

giving us the validity of (3.6) for x ∈ [2n+1, 2n+2

). By the induction principle,

we have shown that formula (3.6) holds true for every n ∈ N0.To prove the converse of (i), we show that, for arbitrary f0 : [1, 2) → (0,∞),

the function f : [1,∞) → (0,∞) defined by (3.4) satisfies (3.3). Indeed, ifx ≥ 1, there exists a unique n ∈ N such that x ∈ [

2n, 2n+1), whence 2x ∈[

2n+1, 2n+2], and the left hand side of (3.3) reads

f (x) = 21+(n−1)2n

x1−2n(f0

( x

2n

))2n

; (3.8)

the right hand side of (3.3) reads

(xf (2x))1/2 =

(

x21+n2n+1(2x)1−2n+1

(f0

(2x

2n+1

))2n+1)1/2

(3.9)

=(

21+n2n+1+1−2n+1x1−2n+1+1

(f0

( x

2n

))2n+1)1/2

= 21+(n−1)2n

x1−2n(f0

( x

2n

))2n

.

Since the right hand sides of (3.8) and (3.9) are equal, the function f definedby (3.4) satisfies (3.3).


(ii) By induction, we shall prove that, for all n ∈ N0,

x ∈[(

12

)n

,

(12

)n−1)

⇒ f (x) = 21−2−n(n+1)x1−2−n(f0

( x

2n+1

))2n

,

(3.10)which is (3.2) with n replaced by −n.

Take arbitrarily x ∈ (0, 1) . There exists a unique n ∈ N0 such that(

12

)n

≤ x <

(12

)n−1

.

If n = 0, then 1 ≤ x < 2 and, by the definition of f0,

f (x) = f0 (x) ,

so (3.10) holds true for n = 0. Assume that (3.10) holds true for some n ∈ N0.

Taking arbitrarily x ∈[(

12

)n+1,(

12

)n)

, we have 2x ∈[(

12

)n,(

12

)n−1), thus,

applying (3.10), we obtain

f (2x) = (2x)1−2−n

21−2−n(n+1)(f0

(2n+1x

))1/2n

.

Hence, by (3.3),

f (x) = (xf (2x))1/2

=(x (2x)1−2−n

21−2−n(n+1)(f0

(2n+1x

))1/2n)1/2

= 21−2−(n+1)(n+2)x1−2−(n+1) (f0

(2n+1x

))1/2n+1

so (3.10) holds true for x ∈[(

12

)n+1,(

12

)n), which means that (3.10) holds for

n + 1. By the induction principle, formula (3.10) holds true for all x ∈ (0, 1).Both reasonings prove the validity of (3.4), which is the second statement ofour theorem.

Arguing similarly as in the previous case we can prove the converse of (ii).In part (iii), since one implication is obvious, we must show that the conti-

nuity of f0 and (3.5) imply the continuity of f . By (3.4), the continuity of f0

on (1, 2) implies that f is continuous on⋃

n∈Z

(2n, 2n+1

). It remains to show the

continuity of f at the point 2n for all n ∈ Z.Applying in turn (3.4), (3.5), we have

limx→2n−

f (x) = limx→2n−

21+(n−2)2n

x1−2n−1(f0

( x

2n−1

))2n−1

= 21+(n−2)2n−n(1−2n−1) limx→2−

(f0 (x))2n−1

= 21+(n−1)2n−n(1−2n)((f0 (1))2

)2n−1

= f (2n) .


The validity of

limx→2n+

f (x) = f (2n)

is obvious since, by (3.4), f is defined on the interval[2n, 2n+1

)and the com-

position and multiplication of continuous functions are continuous.This finishes the proof. �

Remark 5. Since (a,∞) =⋃

n∈N

[a + 1

n ,∞)the counterpart of part (i) holds

true.

From Theorem 1, by Remark 2 and Definition 3, we obtain

Remark 6. A beta-type function Bf is a premean in an interval (a,∞) , a ≥ 0,if, and only if, the function f : (a,∞) → (0,∞) is of the form (3.2).

Remark 7. If f : (0,∞) → (0,∞) is such that (log ◦f)′ ◦ exp is convex and thebeta-type function Bf is a mean in (0,∞), then Bf = H.

Proof. Assume that f : (0,∞) → (0,∞) is a differentiable function. Since Bf

is a mean in (0,∞) , it satisfies (3.3). Taking the logarithm of both sides, wecan write this equation as follows

2 (log ◦f) (x) = (log ◦f) (2x) + log x, x > 0.

Differentiating both sides gives

2 (log ◦f)′ (x) = 2 (log ◦f)′ (2x) +1x

, x > 0,

which can be written in the form

2 (log ◦f)′ ◦ exp (log x) = 2 (log ◦f)′ ◦ exp (log x + log 2) +1

exp (log x), x > 0.

Setting hereh := (log ◦f)′ ◦ exp, u = log x, (3.11)

we conclude that a convex function h satisfies the functional equation

h (u + log 2) = h (u) − e−u

2, u ∈ R. (3.12)

Since the function F : R → R, F (u) := − e−u

2 is concave and

limu→∞ [F (u + log 2) − F (u)] = 0,

Krull’s theorem [2,3] (see also [4], pp. 114–115) gives us the existence of a(unique up to a constant) convex solution h to (3.12). Since, for any realconstant k, the function u −→ e−u + k satisfies the functional equation (3.12)and is convex, it follows that for some k,

h (u) = e−u + k, u ∈ R.


Thus, from (3.11), we get

(log ◦f)′ ◦ exp (u) = e−u + k, u ∈ R.

It follows that

(log ◦f)′ (x) =1x

+ k, x > 0,

whence, setting c = ek, we get

f (x) = bxcx, x > 0,

for some b > 0. �

4. Means in terms of beta-type functions

We use Theorem 1 on the solutions of reflexive beta-type functions to answerthe question when a beta-type function is a mean.

Theorem 2. Let I = (0,∞) or I = [a,∞) for some a > 0 and f : I → (0,∞)be an arbitrary function.

The following conditions are equivalent:(i) the beta-type function Bf : I2 → (0,∞) is a mean;(ii) there is an additive function α : R → R such that

f (x) = 2xeα(x), x ∈ I;

(iii) Bf is the harmonic mean in I.

Proof. To prove the implication (i)=⇒(ii), assume first that I = (0,∞) andBf is a mean in I. Put f0 := f |[1,2) . Since every mean is reflexive, the functionf : (0,∞) → (0,∞) satisfies (3.3). Thus, by part (ii) of Theorem 1 with a = 1,we have, for every n ∈ Z,

x ∈ [2n, 2n+1

) ⇒ f (x) = 21+(n−1)2n

x1−2n(f0

( x

2n

))2n

;

whence, using the definition of beta-type functions,

Bf (x, y) =f (x) f (y)f (x + y)

= 21−2n+1 xy

x + y

[(x + y)2

xy

f0

(x2n

)f0

(y2n

)

(f0

(x+y2n+1

))2

]2n

.

Since Bf is a mean, we have, for all x, y ∈ [2n, 2n+1

),

x ≤ y =⇒ x ≤ 21−2n+1 xy

x + y

[(x + y)2

xy

f0

(x2n

)f0

(y2n

)

(f0

(x+y2n+1

))2

]2n

≤ y.


Choose s, t arbitrarily from [1, 2) . Then, there exists unique n ∈ Z such that

x, y ∈ [2n, 2n+1

)

and

s =x

2nand t =

y

2n.

Since x ≤ y implies s ≤ t, the latter inequality reads

s ≤ 21−2n+1 st

s + t

[(s + t)2

st

f0 (s) f0 (t)(f0

(s+t2

))2

]2n

≤ t, s, t ∈ [1, 2) . (4.1)

The first inequality gives us

s + t

2t≤

[(s + t)2

2st

f0 (s) f0 (t)

2(f0

(s+t2

))2

]2n

, s, t ∈ [1, 2) ,

whence, taking the 2n-th root of both sides,(

s + t

2t

)1/2n

≤ (s + t)2

2st

f0 (s) f0 (t)

2(f0

(s+t2

))2 , s, t ∈ [1, 2) .

Letting n tend to ∞, we obtain

1 ≤ (s + t)2

2st

f0 (s) f0 (t)

2(f0

(s+t2

))2 , s, t ∈ [1, 2) ,

and thus4st

(s + t)2≤ f0 (s) f0 (t)

(f0

(s+t2

))2 , s, t ∈ [1, 2) .

Proceeding analogously, the second inequality in (4.1) gives us

4st

(s + t)2≥ f0 (s) f0 (t)

(f0

(s+t2

))2 , s, t ∈ [1, 2) ;

thus we have4st

(s + t)2=

f0 (s) f0 (t)(f0

(s+t2

))2 , s, t ∈ [1, 2) ,

and consequently(

f0

(s+t2

)

s+t2

)2

=f0 (s)

s

f0 (t)t

, s, t ∈ [1, 2) .

Putting g : [1, 2) → (0,∞) defined by

g (s) =f0 (s)

s,


we obtain(

g

(s + t

2

))2

= g (s) g (t) s, t ∈ [1, 2) .

After taking the logarithm of both sides, we get

2 log g

(s + t

2

)= log g (s) + log g (t) , s, t ∈ [1, 2) ,

thus, dividing both sides by 2, we observe that h := log ◦g satisfies the Jensenequation

h

(s + t

2

)=

h (s) + h (t)2

, s, t ∈ [1, 2) .

By Kuczma [5], p. 351, there exist an additive function α : R → R and k ∈ R

such that

h (t) = α (t) + k, t ∈ [1, 2) .

Substituting back, we thus have

f0 (t) = bteα(t), t ∈ [1, 2) ,

where b := ek. Hence, by (3.4), we have

f (x) = 21−2n

xb2n

eα(x)

and hence the corresponding beta-type function reads

Bf (x, y) =f (x) f (y)f (x + y)

=21−2n

xc2n

eα(x) · 21−2n

yc2n

eα(y)

21−2n+1 (x + y) c2n+1eα(x+y)

= 2xy

x + y

= H (x, y) .

This finishes the proof in the case when I = (0,∞). If I = [a,∞) for somea > 0, applying part (i) of Theorem 1, we can argue similarly. �

In connection with part (ii) of this result, note that very irregular generatorsmay produce quite regular beta-type functions, which is an interesting andcharacteristic phenomenon in the theory of functional equations. This is aconsequence of

Remark 8. (Equality of beta-type functions) We have Bf = Bg if, and only if,there is a function e: R → (0,∞) such that g

f =e|I and e is exponential, i.e.

e (x + y) = e (x) e (y) , x, y ∈ R.

As an application of Theorem 2 and Proposition 1 we get the following


Theorem 3. If a function f : (0,∞) → (0,∞) is logarithmically concave andsatisfies the functional equation

[f (x)]2 = xf (2x) , x > 0,

then there is c > 0 such that

f (x) = 2xcx, x > 0.

Hence, taking Remark 4 into account, we obtain

Corollary 1. If a function f : (0,∞) → (0,∞) is logarithmically convex andsatisfies the functional equation

f (2x) = x [f (x)]2 , x > 0,

then, for some c > 0,

f (x) =12x

cx, x > 0.

5. Final remarks

Our results, formulated for positive-valued functions defined on intervals ofthe form (a,∞) for a ≥ 0, can be transformed to negative-valued functionsdefined on the domain (−∞,−a) , which is shown in the following

Remark 9. Let a ≥ 0. If f : (−∞,−a) → (−∞, 0) satisfies[f (x)

]2 = xf (2x) , x < −a,

then f : (a,∞) → (0,∞) defined by

f (x) = −f (−x) , x > a

satisfies

[f (x)]2 = xf (2x) , x > a,

and thus, by Theorem2 and part (i) of Theorem 1, the function Bf : (−∞,−a)2

→ (−∞, 0) is a mean if, and only if,

Bf (x, y) = −H (−x,−y) , x, y < −a.

From the latter remark it follows that the harmonic mean H : (0,∞)2 →(0,∞) and its negative counterpart H : (−∞, 0)2 → (−∞, 0) defined by

H (x, y) = −H (−x,−y) , x, y < 0,

are means of beta-type. These two means can be “glued” together to obtaina unique increasing mean defined in R

2, which can be treated as a bivariableharmonic mean in R; namely we have the following


Remark 10. The function M : R2→ R defined by

M (x, y) :=

⎧⎨

⎩

H (x, y) if x, y > 0−H (−x,−y) if x, y < 0

0 xy ≤ 0

is the unique increasing one with respect to each variable function such thatM |(0,∞)2 = H and M |(−∞,0)2 = H. Moreover, M is a continuous bivariablemean in R; its restrictions M |

(0,∞)2and M |

(−∞,0)2are of beta-type but, of

course, M is not of beta-type.

Proof. Of course, H and H are increasing. Assume that M : R2 → R is

an increasing function such that M |(0,∞)2

= H and M |(−∞,0)2

= H. Take(x, y) ∈ R

2 such that xy ≤ 0. We consider first the case x ≥ 0, y ≤ 0. Since Mis increasing in both variables, we have

M (x, y) ≥ M (0, y) ≥ limv→y− M (0, v) ≤ lim

u→0−

(lim

v→y− M (u, v))

= limu→0−

(lim

v→y− H (u, v))

= 0,

and

M (x, y) ≤ M (x, 0) ≤ limu→x+

M (u, 0) ≤ limv→0+

(lim

u→x+M (u, v)

)

= limv→0+

(lim

u→x+H (u, v)

)= 0,

implying

M (x, y) = 0, x ≥ 0, y ≤ 0.

The case x ≤ 0, y ≥ 0 is treated analogously yielding

M (x, y) = 0, x ≤ 0, y ≥ 0.

Indeed, the function M is a mean in R since H and H are means in (0,∞)and (−∞, 0) , respectively, and, for xy ≤ 0 with x ≤ y, obviously

x ≤ 0 ≤ y,

holds. The results of the “moreover” part are obvious. �

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References

[1] Himmel, M., Matkowski, J.: Homogeneous beta-type functions. J. Class. Anal. 10(1),59–66 (2017)

[2] Krull, W.: Bemerkungen zur Differenzengleichung g (x + 1) − g (x) = F (x) . I. Math.Nachr. 1, 365–376 (1948)

[3] Krull, W.: Bemerkungen zur Differenzengleichung g (x + 1) − g (x) = F (x) . II. Math.Nachr. 2, 251–262 (1949)

[4] Kuczma, M.: Functional equations in a single variable, Monografie Matematyczne, vol.46, Polish Scientific Publishers, Warszawa (1968)

[5] Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities.

Uniwersytet Slaski, Pan stwowe Wydawnictwo Naukowe, Warszawa-Krakow (1985)[6] Ng, C.T.: Functions generating Schur-convex sums, General Inequalities 5 (Oberwolfach,

1986), pp. 433–438, Internat. Schriftenreihe Numer. Math. 80, Birkhauser, Basel (1987)

Martin Himmel and Janusz MatkowskiFaculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GoraSzafrana 4A65-516 Zielona GoraPolande-mail: [email protected]

Janusz Matkowskie-mail: [email protected]

Received: July 1, 2016

Revised: May 23, 2017

beta-type functions and the harmonic mean · bivariable mean in (a,∞). our main result says that...

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