a new class of analytic functions associated with s gean...

Research ArticleA New Class of Analytic Functions Associated withSslsgean Operator

Muhammad Arif 1 Khurshid Ahmad1 Jin-Lin Liu 2 and Janusz SokoacuteB3

1Department of Mathematics Abdul Wali Khan University Mardan 23200 Mardan Pakistan2Department of Mathematics Yangzhou University Yangzhou 225002 China3University of Rzeszow Faculty of Mathematics and Natural Sciences ul Prof Pigonia 1 35-310 Rzeszow Poland

Correspondence should be addressed to Jin-Lin Liu jlliuyzueducn

Received 29 June 2018 Accepted 15 January 2019 Published 3 February 2019

Academic Editor Mitsuru Sugimoto

Copyright copy 2019 Muhammad Arif et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The main object of the present paper is to investigate a number of useful properties such as inclusion relation distortion boundscoefficient estimates and subordination results for a new subclass of analytic functions which are defined here by means of a linearoperator Several known consequences of the results are also pointed out

1 Introduction and Definitions

LetA denote the set of all functions of the form

119891 (119911) = 119911 + infinsum119896=2

119886119896119911119896 (1)

which are analytic in the open unit diskU = 119911 isin C |119911| lt 1Suppose that 119891 and 119892 are analytic in U then we say that 119891 issubordinate to 119892 written as 119891 ≺ 119892 or 119891(119911) ≺ 119892(119911) if thereexists a function 119908 which is analytic in U with 119908(0) = 0 and|119908(119911)| lt 1 such that

119891 (119911) = 119892 (119908 (119911)) (119911 isin U) (2)

Therefore119891(119911) ≺ 119892(119911) implies119891(U) sub 119892(U) Moreover if thefunction 119892 is univalent in U then the following equivalenceholds

119891 (119911) ≺ 119892 (119911) (119911 isin U) lArrrArr119891 (0) = 119892 (0)

and 119891 (U) sub 119892 (U) (3)

Furthermore letP(119860 119861) minus1 le 119861 lt 119860 le 1 denote the familyof all functions 119902 that are analytic in the open unit diskUwith119902(0) = 1 and satisfy

119902 (119911) ≺ 1 + 1198601199111 + 119861119911 (4)

Geometrically by (3) a function 119902 is in the class P[119860 119861]minus1 le 119861 lt 119860 le 1 if and only if 119902(0) = 1 and 119902(U) sub Ω[119860 119861]where circular domain Ω[119860 119861] is defined by

Ω [119860 119861]=

120596 1003816100381610038161003816100381610038161003816120596 minus 1 minus 1198601198611 minus 1198612 1003816100381610038161003816100381610038161003816 lt 119860 minus 1198611 minus 1198612 for 119861 = minus1120596 Re 120596 gt (1 minus 119860)2 for 119861 = minus1

(5)

ThedomainΩ[119860 119861]119861 = minus1 represents an open circular diskcentered on the real axis with diameter end points1198631 = (1 minus119860)(1 minus 119861) and1198632 = (1 + 119860)(1 + 119861) with 0 lt 1198631 lt 1 lt 1198632

With the help of the class P[119860 119861] we now definethe classes Slowast[119860 119861] and C[119860 119861] of Janowski starlike andJanowski convex functions as below

HindawiJournal of Function SpacesVolume 2019 Article ID 6157394 8 pageshttpsdoiorg10115520196157394

2 Journal of Function Spaces

Slowast [119860 119861] = 119891 isin A 1199111198911015840 (119911)119891 (119911) isin P [119860 119861] (119911 isin U)

C [119860 119861]= 119891 isin A

(1199111198911015840 (119911))10158401198911015840 (119911) isin P [119860 119861] (119911 isin U) (6)

These classes were introduced by Janowski [1] The extensionof Janowski function was discussed by Kuroki Owa andSrivastava [2] by choosing the complex parameters 119860 and 119861with the following conditions

(119894) 119860 = 119861|119861| lt 1|119860| le 1and Re (1 minus 119860119861) ge |119860 minus 119861|

(119894119894) 119860 = 119861|119861| = 1|119860| le 1and Re (1 minus 119860119861) gt 0

(7)

Later on Kuroki and Owa [3] discussed the fact that thecondition |119860| le 1 can be omitted from the conditions in part(119894) of (7) Janowski functions are being studied and extendedin different directions by several renowned mathematicianslike Noor and Arif [4] Arif et al [5] Polatoglu [6] Cho [7]Cho et al [8 9] Liu and Noor [10] Liu and Patel [11] Liu andSrivastava [12 13] etc

For a function 119891 of the form (1) and 119899 = 1 2 3 theSalagean operator [14] is defined by

119863119899119891 (119911) = 119911 + infinsum119896=2

119896119899119886119896119911119896 (8)

It is easy to see that the series119863119899119891(119911) is convergent in the unitdisc for each 119899 isin N Further we have 1198631119891(119911) = 1199111198911015840(119911) Alsowe consider the following differential operator

119863minus1119891 (119911) = int1199110

119891 (120585)120585 d120585 = 119911 + infinsum119896=2

119896minus1119886119896119911119896119863minus119899119891 (119911) = 119863minus1(119863minus(119899minus1)119891 (119911) = 119911 + infinsum

119896=2

119896minus119899119886119896119911119896(9)

for any integers Then for 119891 isin A given by (1) we know that

119863119899119891 (119911) = 119911 + infinsum119896=2

119896119899119886119896119911119896 (119899 = 0 plusmn1 plusmn2 ) (10)

Using this Salagean operator 119863119899along with the concepts ofJanowski functions we now define a subclass ofA as follows

Definition 1 If 119891 isin A then 119891 isin Q119887119904 (119860 119861 119895) if and only if

1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1) ≺ 1 + 1198601199111 + 119861119911 (11)

where minus1 le 119861 lt 119860 le 1 119895 = 1 2 3 sdot sdot sdot and 119887 isin C 0Special Cases In literature various interesting subfamiliesof analytic and univalent functions associated with circulardomain have been studied from a number of different viewpoints which are closely related with the class Q119887119904 (119860 119861 119895)

For example if we set 119895 = 0 in (11) we get the class119876119887119904 (119860 119861 0) equiv Slowast119904 (119887 119860 119861) defined as

Slowast119904 (119887 119860 119861) = 119891 isin A 1 + 1119887 ( 21199111198911015840 (119911)119891 (119911) minus 119891 (minus119911) minus 1)≺ 1 + 1198601199111 + 119861119911 (119911 isin U)

(12)

and further by making 119887 = 2 119860 = 1 119861 = minus1 in Slowast119904 (119887 119860 119861)we obtain the familiar class Slowast119904 of starlike functions withrespect to symmetrical pints studied in [15] Also by putting119895 = 1 in (11) we obtain the familyC119904(119887 119860 119861) defined by

C119904 (119887 119860 119861) = 119891 isin A 1

+ 1119887 ( 2 (1199111198911015840 (119911))1015840(119891 (119911) minus 119891 (minus119911))1015840 minus 1)

≺ 1 + 1198601199111 + 119861119911 (119911 isin U)

(13)

and further taking 119887 = 2 119860 = 1 and 119861 = minus1 in C119904(119887 119860 119861)we get the setC119904 intoduced by Das and Singh [16] For morework see [17ndash21]

We will assume throughout our discussion unless other-wise stated that

minus1 le 119861 lt 119860 le 1and 119887 isin C 0 (14)

2 A Set of Lemmas

Lemma 2 (see [19]) If 119901(119911) = 1 + 1199011119911 + 1199012119911 + sdot sdot sdot belongs tothe classP[119860 119861] then10038161003816100381610038161199011198991003816100381610038161003816 le (119860 minus 119861) forall119899 ge 1 (15)

Lemma 3 (see [19]) Let 119873 be analytic and 119872 starlikefunctions in U with 119873(0) = 119872(0) = 0 Then for minus1 le 119861 lt119860 le 1 100381610038161003816100381610038161198731015840 (119911) 1198721015840 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816119860 minus 119861 (1198731015840 (119911) 1198721015840 (119911))1003816100381610038161003816 lt 1 (16)

Journal of Function Spaces 3

implies

|119873 (119911) 119872 (119911) minus 1||119860 minus 119861 (119873 (119911) 119872 (119911))| lt 1 (119911 isin U) (17)

3 The Main Results and Their Consequences

Theorem 4 If 119891 isin Q119887119904 (119860 119861 119895) then the odd function

Ψ (119911) = 12 [119891 (119911) minus 119891 (minus119911)] (18)

satisfies

1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1) ≺ 1 + 1198601199111 + 119861119911 (19)

Proof If 119891 isin Q119887119904 (119860 119861 119895) then there exists 119901 isin P[119860 119861] suchthat

119887 (119901 (119911) minus 1) = 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1119887 (119901 (minus119911) minus 1) = minus2119863119895+1119891 (minus119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1

(20)

This gives

1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1) = 119901 (119911) + 119901 (minus119911)2 (21)

Because 119901(119911) ≺ (1 + 119860119911)(1 + 119861119911) and (1 + 119860119911)(1 + 119861119911) isunivalent then by (3) we have119901 (119911) + 119901 (minus119911)2 ≺ 1 + 1198601199111 + 119861119911 (22)

It follows (19)

Theorem5 A function119891 isin A is in the classQ119887119904 (119860 119861 119895) if andonly if there exists 119901 isin P[119860 119861] such that

119863119895+1119891 (119911) = 1199112 (1 + 119887(119901 (119911) minus 1)sdot exp[1198872 int

119911

0

(119901 (119905) + 119901 (minus119905) minus 2)119905 d119905] (23)

Proof If 119891 isin Q119887119904 (119860 119861 119895) then it is equivalent to

1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (119911) minus 1)= 1 + 1119887 (119863

119895+1119891 (119911)119863119895Ψ (119911) minus 1) = 119901 (119911)(24)

for some 119901 belonging to the classP[119860 119861] FromTheorem 4we also have

1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1)

= 1 + 1119887 (119911 (119863119895Ψ (119911))1015840119863119895Ψ (119911) minus 1) = 119901 (119911) + 119901 (minus119911)2

(25)

Equation (25) gives

119863119895Ψ (119911) = 119911 exp[1198872 int119911

0

(119901 (119905) + 119901 (minus119905) minus 2)119905 d119905] (26)

and using this in (24) provides (23) It is easy to verify that if119901 belongs to the classP[119860 119861] and 119891 isin A satisfies (23) then119891 isin Q119887119904 (119860 119861 119895)Theorem 6 Let 119891 isin Q119887119904 (119860 119861 119895) Then

100381610038161003816100381611988621003816100381610038161003816 le |119887| (119860 minus 119861)2119895+1 100381610038161003816100381611988631003816100381610038161003816 le |119887| (119860 minus 119861)2 sdot 3119895+1

(27)

And for 119899 ge 2100381610038161003816100381611988621198991003816100381610038161003816 le |119887| (119860 minus 119861)2119895+119899119899119895119899

119899minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896) (28)

and

10038161003816100381610038161198862119899+11003816100381610038161003816 le |119887| (119860 minus 119861)2119899119899 (2119899 + 1)119895119899minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896) (29)

Proof If 119891 isin Q119887119904 (119860 119861 119895) then by Definition 1 we have

1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1) = 1 + 119860119908 (119911)1 + 119861119908 (119911) (30)

Put

119901 (119911) = 1 + infinsum119896=1

119901119896119911119896 = 1 + 119860119908 (119911)1 + 119861119908 (119911) (31)

From (30) we can write

1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1) = 1 +infinsum119896=1

119901119896119911119896 (32)

Also from (32) and (10) we have

119911 + 2119895+111988621199112 + 3119895+111988631199113 + + (2119899)119895+1 11988621198991199112119899 + (2119899+ 1)119895+1 1198862119899+11199112119899+1 + = (119911 + 311989511988631199113 + 511989511988651199115+ + (2119899 minus 1)119895 1198862119899minus11199112119899minus1 + (2119899 + 1)119895 1198862119899+11199112119899+1+ ) times (1 + 119887 (1199011119911 + 11990121199112 + 11990131199113 + + 11990121198991199112119899+ 1199012119899+11199112119899+1 + ))

(33)


Equating the coefficients of like powers of 119911 we have2119895+11198862 = 11988711990112 sdot 31198951198863 = 1198871199012 (34)

4119895+11198864 = 1198871199013 + 3119887119901111988635119895+11198865 = 1198871199014 + 311989511988711990121198863 + 511988651199115 (35)

(2119899)119895+1 1198862119899 = 1198871199012119899minus1 + 311989511988631198871199012119899minus3 + 511989511988651198871199012119899minus5+ + 119887 (2119899 minus 1)119895 1198862119899minus11199011 (36)

(2119899 + 1)119895+1 1198862119899+1 = 1198871199012119899 + 311989511988631198871199012119899minus2 + 511989511988651198871199012119899minus4+ + 119887 (2119899 minus 1)119895 1198862119899minus11199012 (37)

Using Lemma 2 and (34) we easily get


(38)

which gives (27) Using Lemma 2 (27) and (35) we get

100381610038161003816100381611988641003816100381610038161003816 le |119887| (119860 minus 119861)2 sdot 4119895+1 (|119887| (119860 minus 119861) + 2) 100381610038161003816100381611988651003816100381610038161003816 le |119887| (119860 minus 119861)8 sdot 5119895 (|119887| (119860 minus 119861) + 2) (39)

It follows that (28) and (29) hold for 119899 = 2We now prove (28)and (29) by using induction Equation (36) in conjunctionwith Lemma 2 yields

100381610038161003816100381611988621198991003816100381610038161003816 le |119887| (119860 minus 119861)(2119899)119895+1 [1 + 119899minus1sum119903=1

(2119903 + 1)119895 10038161003816100381610038161198862119903+11003816100381610038161003816] (40)

We assume that (28) and (29) hold for 3 4 (119899 minus 1) Thenfrom (40) we obtain


(2119903 + 1)119895 |119887| (119860 minus 119861)2119903119903 (2119903 + 1)119895sdot 119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)] = |119887| (119860 minus 119861)(2119899)119895+1 [1+ 119899minus1sum119903=1

|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)] (41)

where we assumed

0prod119896=1

(|119887| (119860 minus 119861) + 2119896) = 1 (42)

In order to complete the proof it is sufficient to show that

|119887| (119860 minus 119861)(2119904)119895+1 [1+ 119904minus1sum119903=1

|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)]= |119887| (119860 minus 119861)2119895+119904119904119895119904

119904minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896) (119904 = 3 4 )

(43)

Expression (43) is valid for 119904 = 3Let us suppose that (43) is true for 3 (119904minus1)Then from

(41)

|119887| (119860 minus 119861)(2119904)119895+1 [1 + 119904minus1sum119903=1

|119887| (119860 minus 119861)2119903119903sdot 119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)]= (119904 minus 1)119895+1119904119895+1 ( |119887| (119860 minus 119861)(2 (119904 minus 1))119895+1 (1+ 119904minus2sum119903=1

|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)))+ |119887| (119860 minus 119861)(2119904)119895+1 |119887| (119860 minus 119861)2119904minus1 (119904 minus 1)

119904minus2prod119896=1

(|119887| (119860 minus 119861) + 2119896)= (119904 minus 1)119904 |119887| (119860 minus 119861)2119895+119904minus1119904119895 (119904 minus 1)

119904minus2prod119896=1

(|119887| (119860 minus 119861) + 2119896)+ |119887| (119860 minus 119861)2119895+119904minus1119904119895 (119904 minus 1) |119887| (119860 minus 119861)2119904

119904minus2prod119896=1

(|119887| (119860 minus 119861) + 2119896)= |119887| (119860 minus 119861)2119895+119904minus1119904119895 (119904 minus 1)

119904minus2prod119896=1

(|119887| (119860 minus 119861) + 2119896)sdot ( |119887| (119860 minus 119861) + 2 (119904 minus 1)2119904 ) = |119887| (119860 minus 119861)2119895+119904119904119895119904sdot 119904minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)

(44)

Thus (43) holds and hence (41) with (43) implies (28)Similarly we can prove the coefficient estimates given in(29)

Theorem 7 If119863 gt minus1 thenQ119887119904 (119860 119861 119895) sub Q

119887119904 (119862119863 119895) (45)

if and only if1003816100381610038161003816100381610038161003816 1 minus 1198621198631 minus 1198632 minus 1 minus 1198601198611 minus 1198612 1003816100381610038161003816100381610038161003816 le 1 minus 1198621198631 minus 1198632 minus 1 minus 1198601198611 minus 1198612 (46)


If119863 = minus1 thenQ119887119904 (119860 119861 119895) sub Q

119887119904 (119862119863 119895) (47)

if and only if

119862 ge 1 minus 2 (1 minus 119860)1 minus 119861 (48)

Proof For 119891 isin Q119887119904 (119860 119861 119895) we let119867(119891 119911) = 1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1)

119911 isin U(49)

The values of the function119867 lie inΩ[119860 119861] (see (5)) since119891 isin Q119887119904 (119860 119861 119895) lArrrArr

119867(119891U) sub Ω [119860 119861] (50)

Similarly

ℎ isin Q119887119904 (119862119863 119895) lArrrArr119867(119891U) sub Ω [119862119863] (51)

In the case 119861 = minus1 Ω[119860 119861] is a disc 119863(119860 119861) with center119904(119860 119861) and radius 119903(119860 119861)119904 (119860 119861) = 1 minus 1198601198611 minus 1198612 119903 (119860 119861) = 119860 minus 1198611 minus 1198612

(52)

while it is a half plane for 119861 = minus1 Therefore for the case 119861 =minus1 119863 = minus1 the inclusion relation 119876119887119904 (119860 119861 119895) sub 119876119887119904 (119862119863 119895)holds when

119903 (119860 119861) le 119903 (119862119863) (53)

and

|119904 (119862119863) minus 119904 (119860 119861)| le 119903 (119862119863) minus 119903 (119860 119861) (54)

This is equivalent to1003816100381610038161003816100381610038161003816 1 minus 1198621198631 minus 1198632 minus 1 minus 1198601198611 minus 1198612 1003816100381610038161003816100381610038161003816 le 1 minus 1198621198631 minus 1198632 minus 119860 minus 1198611 minus 1198612 (55)

Furthermore we have

Ω [119862 minus1] = 120596 Re 120596 gt (1 minus 119862)2 (56)

The domainΩ[119860 119861] represents an open circular disk or a halfplane on the right site of the point (1 minus119860)(1 minus 119861) Therefore

Ω [119860 119861] sub Ω [119862 minus1] lArrrArr1 minus 1198622 le 1 minus 1198601 minus 119861

(57)

and hence the result follows

Theorem 8 If 119891 isin Q119887119904 (119860 119861 119895) then 119865 isin Q119887119904 (119860 119861 119895) where119865 (119911) = 2119911 int

119911

0119891 (119905) d119905 (58)

Proof From (58) we can easily write

1 + 1119887 ( 2119863119895+1119865 (119911)119863119895119865 (119911) minus 119863119895119865 (minus119911) minus 1)= 2119911119863119895119891 (119911) + (119887 minus 3) int1199110 119863119895119891 (119905) d119905 + (119887 minus 1) int1199110 119863119895119891 (minus119905) d119905119887 [int119911

0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (59)

Let 119873 and119872 be the numerator and denominator functionsrespectively Now we show that

119872(119911) = 119887 [int1199110119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (60)

is starlike Since1198721015840(119911) = 119887[119863119895119891(119911) minus 119863119895119891(minus119911)] therefore1199111198721015840 (119911)119872 (119911) = 119911119863119895119891 (119911) minus 119911119863119895119891 (minus119911)int119911

0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905 (61)

Let

int1199110119863119895119891 (119905) d119905 = ℎ (119911) (62)

Then (61) becomes

1199111198721015840 (119911)119872 (119911) = 12 [ 2119911ℎ1015840 (119911)ℎ (119911) minus ℎ (minus119911) + 2 (minus119911) ℎ1015840 (minus119911)ℎ (minus119911) minus ℎ (119911)] (63)

Since 119891(119911) isin Q119887119904 (119860 119861 119895) it follows that1 + 1119887 ( 2119911ℎ10158401015840 (119911)ℎ1015840 (119911) minus ℎ1015840 (minus119911) minus 1) ≺ 1 + 1198601199111 + 119861119911 (64)

and ℎ(119911) isin C119904(119887 119860 119861) sub Slowast119904 (119887 119860 119861) sub Slowast119904 Now from (63)it follows that119872(119911) is starlike functions Furthermore

1198731015840 (119911)1198721015840 (119911) = 1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1)with 119891 isin Q119887119904 (119860 119861 119895)

(65)

Thus

1198731015840 (119911)1198721015840 (119911) = 1 + 119860119908 (119911)1 + 119861119908 (119911) (66)

This implies that 10038161003816100381610038161003816(1198731015840 (119911) 1198721015840 (119911) minus 1)100381610038161003816100381610038161003816100381610038161003816(119860 minus 119861 (1198731015840 (119911) 1198721015840 (119911)))1003816100381610038161003816 lt 1 (67)

By Lemma 3 we have

|(119873 (119911) 119872 (119911) minus 1)||(119860 minus 119861 (119873 (119911) 119872 (119911)))| lt 1 119911 isin U (68)

and this gives that 119865 isin Q119887119904 (119860 119861 119895)


Theorem 9 If 119865 isin Q119887119904 (1 minus1 119895) in |119911| lt 1 then 119891(119911) =(12)(119911119865(119911))1015840 belongs to the classQ119887119904 (1 minus1 119895) in |119911| lt 1199030 where1199030 = 2

1 + |119887| + radic|119887|2 minus 2 |119887| + 5 (69)

Proof Since 119865 isin Q119887119904 (1 minus1 119895) it follows thatRe(1 + 1119887 ( 2119863119895+1119865 (119911)119863119895119865 (119911) minus 119863119895119865 (minus119911) minus 1)) gt 0

119911 isin U(70)

Equivalently we have

1 + 1119887 ( 2119863119895+1119865 (119911)119863119895119865 (119911) minus 119863119895119865 (minus119911) minus 1) = 1 minus 119908 (119911)1 + 119908 (119911) (71)

where 119908 is analytic in U with |119908(119911)| lt 1 and 119908(0) = 0After simple computation we obtain


0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (72)

Using (71) and (72) we can show that

1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1) = 1 minus 119908 (119911)1 + 119908 (119911) minus 21199111199081015840 (119911) (1 + 119908 (minus119911))(2 minus 119887) (1 + 119908 (119911))2 (1 + 119908 (minus119911)) + 119887 (1 minus 119908 (119911)119908 (minus119911)) (1 + 119908 (119911)) (73)

Thus 119891 isin Q119887119904 (1 minus1 119895) if

Re(1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1))= Re[1 minus 119908 (119911)1 + 119908 (119911) minus 21199111199081015840 (119911) (1 + 119908 (minus119911))(2 minus 119887) (1 + 119908 (119911))2 (1 + 119908 (minus119911)) + 119887 (1 minus 119908 (119911)119908 (minus119911)) (1 + 119908 (119911))] gt 0 in |119911| lt 1199030

(74)

Now

Re(1 minus 119908 (119911)1 + 119908 (119911)) = 1 minus |119908 (119911)|2

1 + |119908 (119911)|2 (75)

and

Re[ 21199111199081015840 (119911) (1 + 119908 (minus119911))(2 minus 119887) (1 + 119908 (119911))2 (1 + 119908 (minus119911)) + 119887 (1 minus 119908 (119911)119908 (minus119911)) (1 + 119908 (119911))]le 2 |119911| |1 + 119908 (minus119911)|10038161003816100381610038161003816(2 minus 119887) (1 + 119908 (119911))2 (1 + 119908 (minus119911))10038161003816100381610038161003816 + |119887 (1 minus 119908 (119911)119908 (minus119911)) (1 + 119908 (119911))| times

1 minus |119908 (119911)|21 minus |119911|2 (76)

where we have used the well-known estimate100381610038161003816100381610038161199081015840 (119911)10038161003816100381610038161003816 le 1 minus |119908 (119911)|21 minus |119911|2 for |119911| lt 1 (77)

Therefore (74) is true if1|119887| ( 2 |119911|1 minus |119911|2 minus (2 minus |119887|)) + 1lt |2 + 119908 (119911) + 119908 (minus119911)||1 + 119908 (119911)| |1 + 119908 (minus119911)| = 1003816100381610038161003816ℎ1 (119911) + ℎ2 (119911)1003816100381610038161003816

(78)

where

ℎ1 (119911) = 11 + 119908 (119911)and ℎ2 (119911) = 11 + 119908 (minus119911)

(79)

Since

1 minus 119903 le |1 + 119908 (119911)| le 1 + 119903 (80)


this implies that

11 + 119903 le ℎ119894 (119911) le 11 minus 119903 for 119894 = 1 2 (81)

Therefore we have

1003816100381610038161003816ℎ1 (119911) + ℎ2 (119911)1003816100381610038161003816 ge 21 + 119903 (82)

and from (78) it follows that 119891 isin Q119887119904 (119860 119861 119895) if1199031 minus 1199032 minus (1 minus |119887|) lt |119887|1 + 119903 (83)

or

119903 lt 2(1 + |119887|) + radic|119887|2 minus 2 |119887| + 5 (84)

Thus (74) is true if the last inequality holds

Theorem 10 Let 119891 isin 119876119887119904 (119860 119861 119895) Then for |119911| = 119903 0 lt 119903 lt 1(1 minus |119887| 119860119903) 119903 + (1 minus |119887|) 1198611199032(1 + 1199032) (1 minus 119861119903) le 10038161003816100381610038161003816119863119895+1119891 (119911)10038161003816100381610038161003816le (1 + |119887| 119860119903) 119903 + (1 minus |119887|) 1198611199032(1 minus 1199032) (1 + 119861119903)

(85)

Proof Let us put ℎ(119911) = (119863119895119891(119911) minus 119863119895119891(minus119911))2 Then weobtain

10038161003816100381610038161003816119863119895+1119891 (119911)10038161003816100381610038161003816 = |ℎ (119911)| 100381610038161003816100381610038161003816100381610038161 + 119887 (119860 minus 119861)119908 (119911)1 + 119861119908 (119911)10038161003816100381610038161003816100381610038161003816 (86)

Since ℎ is odd starlike it follows that119903(1 + 1199032) le |ℎ (119911)| le 119903(1 minus 1199032) (87)

Furthermore for 119908 isin A one can easily obtain that

(1 minus |119887| 119860119903) + (1 minus |119887|) 1198611199031 minus 119861119903le 10038161003816100381610038161003816100381610038161003816 (1 + 119861119908 (119911) + 119887 (119860 minus 119861)119908 (119911)1 + 119861119908 (119911)

10038161003816100381610038161003816100381610038161003816le (1 + |119887| 119860119903) + (1 minus |119887|) 1198611199031 + 119861119903

(88)

Applying the last inequalities along with (87) in (86) we easilyobtain (85)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors agree with the contents of the manuscript andthere are no conflicts of interest among the authors

Acknowledgments

This work is supported by National Natural Science Founda-tion of China (Grant no 11571299) andNatural Science Foun-dation of Jiangsu Province of China (Grant no BK20151304)

References

[1] W Janowski ldquoSome extremal problems for certain families ofanalytic functionsrdquo Annales Polonici Mathematici vol 28 pp297ndash326 1973

[2] K Kuroki S Owa and H M Srivastava ldquoSome subordinationcriteria for analytic functionsrdquo Bulletin de la Societe des Scienceset des Lettres de Łodz vol 52 pp 27ndash36 2007

[3] K Kuroki and S Owa ldquoSome subordination criteria concerningthe Salagean operatorrdquo Journal of Inequalities in Pure andApplied Mathematics vol 10 no 2 Article 36 11 pages 2009

[4] K I Noor and M Arif ldquoMapping properties of an integraloperatorrdquo Applied Mathematics Letters vol 25 no 11 pp 1826ndash1829 2012

[5] M Arif K I Noor M Raza and W Haq ldquoSome properties ofa generalized class of analytic functions related with janowskifunctionsrdquo Abstract and Applied Analysis vol 2012 Article ID279843 11 pages 2012

[6] Y Polatoglu M Bolcal A Sen and E Yavuz ldquoA study onthe generalization of Janowski functions in the unit discrdquo ActaMathematica Academiae Paedagogiace Nyıregyhaziensis vol 22no 1 pp 27ndash31 2006

[7] N E Cho and I H Kim ldquoInclusion properties of certainclasses of meromorphic functions associated with the gen-eralized hypergeometric functionrdquo Applied Mathematics andComputation vol 187 no 1 pp 115ndash121 2007

[8] N E Cho ldquoThe Noor integral operator and strongly close-to-convex functionsrdquo Journal of Mathematical Analysis andApplications vol 283 no 1 pp 202ndash212 2003

[9] N E Cho O S Kwon and H M Srivastava ldquoInclusionrelationships and argument properties for certain subclassesof multivalent functions associated with a family of linearoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 292 no 2 pp 470ndash483 2004

[10] J-L Liu and K I Noor ldquoOn subordinations for certain analyticfunctions associated with Noor integral operatorrdquo AppliedMathematics and Computation vol 187 no 2 pp 1453ndash14602007

[11] J-L Liu and J Patel ldquoCertain properties of multivalent func-tions associated with an extended fractional differintegraloperatorrdquo Applied Mathematics and Computation vol 203 no2 pp 703ndash713 2008

[12] J Liu and H M Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001

[13] J Liu and H M Srivastava ldquoCertain properties of the Dziok-Srivastava operatorrdquo Applied Mathematics and Computationvol 159 no 2 pp 485ndash493 2004

[14] G S Salagean ldquoSubclasses of univalent functionsrdquo in Proceed-ings of the Complex Analysis 5th Romanian Finnish Seminarvol 1013 of Lecture Notes inMathematics pp 362ndash372 SpringerBucharest Romania 1983

[15] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959


[16] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977

[17] C-Y Gao and S-Q Zhou ldquoOn a class of analytic functionsrelated to the starlike functionsrdquo Kyungpook MathematicalJournal vol 45 no 1 pp 123ndash130 2005

[18] M Arif Z-G Wang R Khan and S K Lee ldquoCoefficientinequalities for Janowski-Sakaguchi type functions associatedwith conic regionsrdquo Hacettepe Journal of Mathematics andStatistics vol 47 no 2 pp 261ndash271 2018

[19] R M Goel and B S Mehrok ldquoA subclass of starlike functionswith respect to symmetric pointsrdquo Tamkang Journal of Mathe-matics vol 13 no 1 pp 11ndash24 1982

[20] H M Srivastava M R Khan and M Arif ldquoSome subclassesof close-to-convex mappings associated with conic regionsrdquoApplied Mathematics and Computation vol 285 pp 94ndash1022016

[21] J Thangamani ldquoThe radius of univalence of certain analyticfunctionsrdquo Indian Journal of Pure and AppliedMathematics vol10 no 11 pp 1369ndash1373 1979

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Slowast [119860 119861] = 119891 isin A 1199111198911015840 (119911)119891 (119911) isin P [119860 119861] (119911 isin U)

C [119860 119861]= 119891 isin A

(1199111198911015840 (119911))10158401198911015840 (119911) isin P [119860 119861] (119911 isin U) (6)

These classes were introduced by Janowski [1] The extensionof Janowski function was discussed by Kuroki Owa andSrivastava [2] by choosing the complex parameters 119860 and 119861with the following conditions

(119894) 119860 = 119861|119861| lt 1|119860| le 1and Re (1 minus 119860119861) ge |119860 minus 119861|

(119894119894) 119860 = 119861|119861| = 1|119860| le 1and Re (1 minus 119860119861) gt 0

(7)

Later on Kuroki and Owa [3] discussed the fact that thecondition |119860| le 1 can be omitted from the conditions in part(119894) of (7) Janowski functions are being studied and extendedin different directions by several renowned mathematicianslike Noor and Arif [4] Arif et al [5] Polatoglu [6] Cho [7]Cho et al [8 9] Liu and Noor [10] Liu and Patel [11] Liu andSrivastava [12 13] etc

For a function 119891 of the form (1) and 119899 = 1 2 3 theSalagean operator [14] is defined by

119863119899119891 (119911) = 119911 + infinsum119896=2

119896119899119886119896119911119896 (8)

It is easy to see that the series119863119899119891(119911) is convergent in the unitdisc for each 119899 isin N Further we have 1198631119891(119911) = 1199111198911015840(119911) Alsowe consider the following differential operator

119863minus1119891 (119911) = int1199110

119891 (120585)120585 d120585 = 119911 + infinsum119896=2

119896minus1119886119896119911119896119863minus119899119891 (119911) = 119863minus1(119863minus(119899minus1)119891 (119911) = 119911 + infinsum

119896=2

119896minus119899119886119896119911119896(9)

for any integers Then for 119891 isin A given by (1) we know that

119863119899119891 (119911) = 119911 + infinsum119896=2

119896119899119886119896119911119896 (119899 = 0 plusmn1 plusmn2 ) (10)

Using this Salagean operator 119863119899along with the concepts ofJanowski functions we now define a subclass ofA as follows

Definition 1 If 119891 isin A then 119891 isin Q119887119904 (119860 119861 119895) if and only if

1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (minus119911) minus 1) ≺ 1 + 1198601199111 + 119861119911 (11)

where minus1 le 119861 lt 119860 le 1 119895 = 1 2 3 sdot sdot sdot and 119887 isin C 0Special Cases In literature various interesting subfamiliesof analytic and univalent functions associated with circulardomain have been studied from a number of different viewpoints which are closely related with the class Q119887119904 (119860 119861 119895)

For example if we set 119895 = 0 in (11) we get the class119876119887119904 (119860 119861 0) equiv Slowast119904 (119887 119860 119861) defined as

Slowast119904 (119887 119860 119861) = 119891 isin A 1 + 1119887 ( 21199111198911015840 (119911)119891 (119911) minus 119891 (minus119911) minus 1)≺ 1 + 1198601199111 + 119861119911 (119911 isin U)

(12)

and further by making 119887 = 2 119860 = 1 119861 = minus1 in Slowast119904 (119887 119860 119861)we obtain the familiar class Slowast119904 of starlike functions withrespect to symmetrical pints studied in [15] Also by putting119895 = 1 in (11) we obtain the familyC119904(119887 119860 119861) defined by

C119904 (119887 119860 119861) = 119891 isin A 1

+ 1119887 ( 2 (1199111198911015840 (119911))1015840(119891 (119911) minus 119891 (minus119911))1015840 minus 1)

≺ 1 + 1198601199111 + 119861119911 (119911 isin U)

(13)

and further taking 119887 = 2 119860 = 1 and 119861 = minus1 in C119904(119887 119860 119861)we get the setC119904 intoduced by Das and Singh [16] For morework see [17ndash21]

We will assume throughout our discussion unless other-wise stated that

minus1 le 119861 lt 119860 le 1and 119887 isin C 0 (14)

2 A Set of Lemmas

Lemma 2 (see [19]) If 119901(119911) = 1 + 1199011119911 + 1199012119911 + sdot sdot sdot belongs tothe classP[119860 119861] then10038161003816100381610038161199011198991003816100381610038161003816 le (119860 minus 119861) forall119899 ge 1 (15)

Lemma 3 (see [19]) Let 119873 be analytic and 119872 starlikefunctions in U with 119873(0) = 119872(0) = 0 Then for minus1 le 119861 lt119860 le 1 100381610038161003816100381610038161198731015840 (119911) 1198721015840 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816119860 minus 119861 (1198731015840 (119911) 1198721015840 (119911))1003816100381610038161003816 lt 1 (16)


implies




Ψ (119911) = 12 [119891 (119911) minus 119891 (minus119911)] (18)

satisfies

1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1) ≺ 1 + 1198601199111 + 119861119911 (19)



(20)

This gives

1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1) = 119901 (119911) + 119901 (minus119911)2 (21)


It follows (19)



119911

0

(119901 (119905) + 119901 (minus119905) minus 2)119905 d119905] (23)


1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (119911) minus 1)= 1 + 1119887 (119863

119895+1119891 (119911)119863119895Ψ (119911) minus 1) = 119901 (119911)(24)


1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1)

= 1 + 1119887 (119911 (119863119895Ψ (119911))1015840119863119895Ψ (119911) minus 1) = 119901 (119911) + 119901 (minus119911)2

(25)

Equation (25) gives

119863119895Ψ (119911) = 119911 exp[1198872 int119911

0

(119901 (119905) + 119901 (minus119905) minus 2)119905 d119905] (26)



(27)


119899minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896) (28)

and


(|119887| (119860 minus 119861) + 2119896) (29)



Put

119901 (119911) = 1 + infinsum119896=1

119901119896119911119896 = 1 + 119860119908 (119911)1 + 119861119908 (119911) (31)



119901119896119911119896 (32)



(33)



4119895+11198864 = 1198871199013 + 3119887119901111988635119895+11198865 = 1198871199014 + 311989511988711990121198863 + 511988651199115 (35)





(38)





(2119903 + 1)119895 10038161003816100381610038161198862119903+11003816100381610038161003816] (40)





|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)] (41)

where we assumed

0prod119896=1

(|119887| (119860 minus 119861) + 2119896) = 1 (42)


|119887| (119860 minus 119861)(2119904)119895+1 [1+ 119904minus1sum119903=1

|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)]= |119887| (119860 minus 119861)2119895+119904119904119895119904

119904minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896) (119904 = 3 4 )

(43)


(41)

|119887| (119860 minus 119861)(2119904)119895+1 [1 + 119904minus1sum119903=1



|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


(|119887| (119860 minus 119861) + 2119896)

(44)



119887119904 (119862119863 119895) (45)




119887119904 (119862119863 119895) (47)

if and only if



119911 isin U(49)


119867(119891U) sub Ω [119860 119861] (50)

Similarly



(52)


119903 (119860 119861) le 119903 (119862119863) (53)

and

|119904 (119862119863) minus 119904 (119860 119861)| le 119903 (119862119863) minus 119903 (119860 119861) (54)


Furthermore we have




(57)



119911

0119891 (119905) d119905 (58)



0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (59)


119872(119911) = 119887 [int1199110119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (60)


0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905 (61)

Let

int1199110119863119895119891 (119905) d119905 = ℎ (119911) (62)

Then (61) becomes





(65)

Thus

1198731015840 (119911)1198721015840 (119911) = 1 + 119860119908 (119911)1 + 119861119908 (119911) (66)


By Lemma 3 we have





1 + |119887| + radic|119887|2 minus 2 |119887| + 5 (69)


119911 isin U(70)





0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (72)





(74)

Now

Re(1 minus 119908 (119911)1 + 119908 (119911)) = 1 minus |119908 (119911)|2

1 + |119908 (119911)|2 (75)

and


1 minus |119908 (119911)|21 minus |119911|2 (76)



(78)

where

ℎ1 (119911) = 11 + 119908 (119911)and ℎ2 (119911) = 11 + 119908 (minus119911)

(79)

Since

1 minus 119903 le |1 + 119908 (119911)| le 1 + 119903 (80)


this implies that

11 + 119903 le ℎ119894 (119911) le 11 minus 119903 for 119894 = 1 2 (81)

Therefore we have

1003816100381610038161003816ℎ1 (119911) + ℎ2 (119911)1003816100381610038161003816 ge 21 + 119903 (82)


or

119903 lt 2(1 + |119887|) + radic|119887|2 minus 2 |119887| + 5 (84)



(85)


10038161003816100381610038161003816119863119895+1119891 (119911)10038161003816100381610038161003816 = |ℎ (119911)| 100381610038161003816100381610038161003816100381610038161 + 119887 (119860 minus 119861)119908 (119911)1 + 119861119908 (119911)10038161003816100381610038161003816100381610038161003816 (86)




10038161003816100381610038161003816100381610038161003816le (1 + |119887| 119860119903) + (1 minus |119887|) 1198611199031 + 119861119903

(88)


Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018









implies




Ψ (119911) = 12 [119891 (119911) minus 119891 (minus119911)] (18)

satisfies

1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1) ≺ 1 + 1198601199111 + 119861119911 (19)



(20)

This gives

1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1) = 119901 (119911) + 119901 (minus119911)2 (21)


It follows (19)



119911

0

(119901 (119905) + 119901 (minus119905) minus 2)119905 d119905] (23)


1 + 1119887 ( 2119863119895+1119891 (119911)119863119895119891 (119911) minus 119863119895119891 (119911) minus 1)= 1 + 1119887 (119863

119895+1119891 (119911)119863119895Ψ (119911) minus 1) = 119901 (119911)(24)


1 + 1119887 (119863119895+1Ψ (119911)119863119895Ψ (119911) minus 1)

= 1 + 1119887 (119911 (119863119895Ψ (119911))1015840119863119895Ψ (119911) minus 1) = 119901 (119911) + 119901 (minus119911)2

(25)

Equation (25) gives

119863119895Ψ (119911) = 119911 exp[1198872 int119911

0

(119901 (119905) + 119901 (minus119905) minus 2)119905 d119905] (26)



(27)


119899minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896) (28)

and


(|119887| (119860 minus 119861) + 2119896) (29)



Put

119901 (119911) = 1 + infinsum119896=1

119901119896119911119896 = 1 + 119860119908 (119911)1 + 119861119908 (119911) (31)



119901119896119911119896 (32)



(33)



4119895+11198864 = 1198871199013 + 3119887119901111988635119895+11198865 = 1198871199014 + 311989511988711990121198863 + 511988651199115 (35)





(38)





(2119903 + 1)119895 10038161003816100381610038161198862119903+11003816100381610038161003816] (40)





|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)] (41)

where we assumed

0prod119896=1

(|119887| (119860 minus 119861) + 2119896) = 1 (42)


|119887| (119860 minus 119861)(2119904)119895+1 [1+ 119904minus1sum119903=1

|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)]= |119887| (119860 minus 119861)2119895+119904119904119895119904

119904minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896) (119904 = 3 4 )

(43)


(41)

|119887| (119860 minus 119861)(2119904)119895+1 [1 + 119904minus1sum119903=1



|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


(|119887| (119860 minus 119861) + 2119896)

(44)



119887119904 (119862119863 119895) (45)




119887119904 (119862119863 119895) (47)

if and only if



119911 isin U(49)


119867(119891U) sub Ω [119860 119861] (50)

Similarly



(52)


119903 (119860 119861) le 119903 (119862119863) (53)

and

|119904 (119862119863) minus 119904 (119860 119861)| le 119903 (119862119863) minus 119903 (119860 119861) (54)


Furthermore we have




(57)



119911

0119891 (119905) d119905 (58)



0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (59)


119872(119911) = 119887 [int1199110119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (60)


0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905 (61)

Let

int1199110119863119895119891 (119905) d119905 = ℎ (119911) (62)

Then (61) becomes





(65)

Thus

1198731015840 (119911)1198721015840 (119911) = 1 + 119860119908 (119911)1 + 119861119908 (119911) (66)


By Lemma 3 we have





1 + |119887| + radic|119887|2 minus 2 |119887| + 5 (69)


119911 isin U(70)





0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (72)





(74)

Now

Re(1 minus 119908 (119911)1 + 119908 (119911)) = 1 minus |119908 (119911)|2

1 + |119908 (119911)|2 (75)

and


1 minus |119908 (119911)|21 minus |119911|2 (76)



(78)

where

ℎ1 (119911) = 11 + 119908 (119911)and ℎ2 (119911) = 11 + 119908 (minus119911)

(79)

Since

1 minus 119903 le |1 + 119908 (119911)| le 1 + 119903 (80)


this implies that

11 + 119903 le ℎ119894 (119911) le 11 minus 119903 for 119894 = 1 2 (81)

Therefore we have

1003816100381610038161003816ℎ1 (119911) + ℎ2 (119911)1003816100381610038161003816 ge 21 + 119903 (82)


or

119903 lt 2(1 + |119887|) + radic|119887|2 minus 2 |119887| + 5 (84)



(85)


10038161003816100381610038161003816119863119895+1119891 (119911)10038161003816100381610038161003816 = |ℎ (119911)| 100381610038161003816100381610038161003816100381610038161 + 119887 (119860 minus 119861)119908 (119911)1 + 119861119908 (119911)10038161003816100381610038161003816100381610038161003816 (86)




10038161003816100381610038161003816100381610038161003816le (1 + |119887| 119860119903) + (1 minus |119887|) 1198611199031 + 119861119903

(88)


Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018










4119895+11198864 = 1198871199013 + 3119887119901111988635119895+11198865 = 1198871199014 + 311989511988711990121198863 + 511988651199115 (35)





(38)





(2119903 + 1)119895 10038161003816100381610038161198862119903+11003816100381610038161003816] (40)





|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)] (41)

where we assumed

0prod119896=1

(|119887| (119860 minus 119861) + 2119896) = 1 (42)


|119887| (119860 minus 119861)(2119904)119895+1 [1+ 119904minus1sum119903=1

|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896)]= |119887| (119860 minus 119861)2119895+119904119904119895119904

119904minus1prod119896=1

(|119887| (119860 minus 119861) + 2119896) (119904 = 3 4 )

(43)


(41)

|119887| (119860 minus 119861)(2119904)119895+1 [1 + 119904minus1sum119903=1



|119887| (119860 minus 119861)2119903119903119903minus1prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


119904minus2prod119896=1


(|119887| (119860 minus 119861) + 2119896)

(44)



119887119904 (119862119863 119895) (45)




119887119904 (119862119863 119895) (47)

if and only if



119911 isin U(49)


119867(119891U) sub Ω [119860 119861] (50)

Similarly



(52)


119903 (119860 119861) le 119903 (119862119863) (53)

and

|119904 (119862119863) minus 119904 (119860 119861)| le 119903 (119862119863) minus 119903 (119860 119861) (54)


Furthermore we have




(57)



119911

0119891 (119905) d119905 (58)



0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (59)


119872(119911) = 119887 [int1199110119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (60)


0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905 (61)

Let

int1199110119863119895119891 (119905) d119905 = ℎ (119911) (62)

Then (61) becomes





(65)

Thus

1198731015840 (119911)1198721015840 (119911) = 1 + 119860119908 (119911)1 + 119861119908 (119911) (66)


By Lemma 3 we have





1 + |119887| + radic|119887|2 minus 2 |119887| + 5 (69)


119911 isin U(70)





0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (72)





(74)

Now

Re(1 minus 119908 (119911)1 + 119908 (119911)) = 1 minus |119908 (119911)|2

1 + |119908 (119911)|2 (75)

and


1 minus |119908 (119911)|21 minus |119911|2 (76)



(78)

where

ℎ1 (119911) = 11 + 119908 (119911)and ℎ2 (119911) = 11 + 119908 (minus119911)

(79)

Since

1 minus 119903 le |1 + 119908 (119911)| le 1 + 119903 (80)


this implies that

11 + 119903 le ℎ119894 (119911) le 11 minus 119903 for 119894 = 1 2 (81)

Therefore we have

1003816100381610038161003816ℎ1 (119911) + ℎ2 (119911)1003816100381610038161003816 ge 21 + 119903 (82)


or

119903 lt 2(1 + |119887|) + radic|119887|2 minus 2 |119887| + 5 (84)



(85)


10038161003816100381610038161003816119863119895+1119891 (119911)10038161003816100381610038161003816 = |ℎ (119911)| 100381610038161003816100381610038161003816100381610038161 + 119887 (119860 minus 119861)119908 (119911)1 + 119861119908 (119911)10038161003816100381610038161003816100381610038161003816 (86)




10038161003816100381610038161003816100381610038161003816le (1 + |119887| 119860119903) + (1 minus |119887|) 1198611199031 + 119861119903

(88)


Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018










119887119904 (119862119863 119895) (47)

if and only if



119911 isin U(49)


119867(119891U) sub Ω [119860 119861] (50)

Similarly



(52)


119903 (119860 119861) le 119903 (119862119863) (53)

and

|119904 (119862119863) minus 119904 (119860 119861)| le 119903 (119862119863) minus 119903 (119860 119861) (54)


Furthermore we have




(57)



119911

0119891 (119905) d119905 (58)



0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (59)


119872(119911) = 119887 [int1199110119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (60)


0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905 (61)

Let

int1199110119863119895119891 (119905) d119905 = ℎ (119911) (62)

Then (61) becomes





(65)

Thus

1198731015840 (119911)1198721015840 (119911) = 1 + 119860119908 (119911)1 + 119861119908 (119911) (66)


By Lemma 3 we have





1 + |119887| + radic|119887|2 minus 2 |119887| + 5 (69)


119911 isin U(70)





0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (72)





(74)

Now

Re(1 minus 119908 (119911)1 + 119908 (119911)) = 1 minus |119908 (119911)|2

1 + |119908 (119911)|2 (75)

and


1 minus |119908 (119911)|21 minus |119911|2 (76)



(78)

where

ℎ1 (119911) = 11 + 119908 (119911)and ℎ2 (119911) = 11 + 119908 (minus119911)

(79)

Since

1 minus 119903 le |1 + 119908 (119911)| le 1 + 119903 (80)


this implies that

11 + 119903 le ℎ119894 (119911) le 11 minus 119903 for 119894 = 1 2 (81)

Therefore we have

1003816100381610038161003816ℎ1 (119911) + ℎ2 (119911)1003816100381610038161003816 ge 21 + 119903 (82)


or

119903 lt 2(1 + |119887|) + radic|119887|2 minus 2 |119887| + 5 (84)



(85)


10038161003816100381610038161003816119863119895+1119891 (119911)10038161003816100381610038161003816 = |ℎ (119911)| 100381610038161003816100381610038161003816100381610038161 + 119887 (119860 minus 119861)119908 (119911)1 + 119861119908 (119911)10038161003816100381610038161003816100381610038161003816 (86)




10038161003816100381610038161003816100381610038161003816le (1 + |119887| 119860119903) + (1 minus |119887|) 1198611199031 + 119861119903

(88)


Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018










1 + |119887| + radic|119887|2 minus 2 |119887| + 5 (69)


119911 isin U(70)





0119863119895119891 (119905) d119905 + int119911

0119863119895119891 (minus119905) d119905] (72)





(74)

Now

Re(1 minus 119908 (119911)1 + 119908 (119911)) = 1 minus |119908 (119911)|2

1 + |119908 (119911)|2 (75)

and


1 minus |119908 (119911)|21 minus |119911|2 (76)



(78)

where

ℎ1 (119911) = 11 + 119908 (119911)and ℎ2 (119911) = 11 + 119908 (minus119911)

(79)

Since

1 minus 119903 le |1 + 119908 (119911)| le 1 + 119903 (80)


this implies that

11 + 119903 le ℎ119894 (119911) le 11 minus 119903 for 119894 = 1 2 (81)

Therefore we have

1003816100381610038161003816ℎ1 (119911) + ℎ2 (119911)1003816100381610038161003816 ge 21 + 119903 (82)


or

119903 lt 2(1 + |119887|) + radic|119887|2 minus 2 |119887| + 5 (84)



(85)


10038161003816100381610038161003816119863119895+1119891 (119911)10038161003816100381610038161003816 = |ℎ (119911)| 100381610038161003816100381610038161003816100381610038161 + 119887 (119860 minus 119861)119908 (119911)1 + 119861119908 (119911)10038161003816100381610038161003816100381610038161003816 (86)




10038161003816100381610038161003816100381610038161003816le (1 + |119887| 119860119903) + (1 minus |119887|) 1198611199031 + 119861119903

(88)


Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018









this implies that

11 + 119903 le ℎ119894 (119911) le 11 minus 119903 for 119894 = 1 2 (81)

Therefore we have

1003816100381610038161003816ℎ1 (119911) + ℎ2 (119911)1003816100381610038161003816 ge 21 + 119903 (82)


or

119903 lt 2(1 + |119887|) + radic|119887|2 minus 2 |119887| + 5 (84)



(85)


10038161003816100381610038161003816119863119895+1119891 (119911)10038161003816100381610038161003816 = |ℎ (119911)| 100381610038161003816100381610038161003816100381610038161 + 119887 (119860 minus 119861)119908 (119911)1 + 119861119908 (119911)10038161003816100381610038161003816100381610038161003816 (86)




10038161003816100381610038161003816100381610038161003816le (1 + |119887| 119860119903) + (1 minus |119887|) 1198611199031 + 119861119903

(88)


Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018






















Journal of












Journal of







Volume 2018






Volume 2018








a new class of analytic functions associated with s gean...

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