Research Article Convexity Properties for Certain Classes of … · 2019. 7. 31. · Moreover, Frasin [ ] noticed that for suitable functions % A ,theintegraloperator3 generalizes

Research ArticleConvexity Properties for Certain Classes of Analytic FunctionsAssociated with an Integral Operator

Ben Wongsaijai and Nattakorn Sukantamala

Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand

Correspondence should be addressed to Nattakorn Sukantamala gaia2556gmailcom

Received 30 May 2014 Revised 4 September 2014 Accepted 4 September 2014 Published 14 October 2014

Academic Editor Valery Serov

Copyright copy 2014 B Wongsaijai and N Sukantamala This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We introduce a new form of generalized integral operator defined on the class of analytic functions A0 By making use of this

novel integral operator we give the convexity of other integral operators We also briefly indicate the relevant connections of ourpresented results to the formerly reported results Furthermore other interesting properties are also discussed

1 Introduction

In the field of geometric function theory the class of univalentfunctions [1 2] has been mainly studied There are manydistinguished geometric properties that played importantrole in the theory of univalent functions such as starlikenessconvexity and close-to-convexity (see eg [3ndash5]) One ofthe generalizations of univalent functions is the theoryof multivalent functions or 119901-valent functions Also thegeometric properties for the subclasses of 119901-valent functionsare investigated by many authors (see eg [6ndash9])

Let D = 119911 isin C |119911| lt 1 be an open unit disc in thecomplex plane For a positive integer 119901A

119901denotes the class

of 119901-valent functions of the following form

119891 (119911) = 119911119901+

infin

sum

119895=1

119886119901+119895119911119901+119895 (1)

which is analytic in D In particular we set A1equiv A

Furthermore let A0be the class of analytic functions 119891 in

D of the following form

119891 (119911) = 1 +

infin

sum

119895=1

119887119895119911119895 (2)

A function 119891 isin A119901is said to be 119901-valently starlike of

order 120574 (0 le 120574 lt 119901) in D if 119891 satisfies

Re1199111198911015840(119911)

119891 (119911) gt 120574 119911 isin D (3)

We denote this class by Slowast119901(120574) which was introduced by Patil

andThakare [10] In particular we set Slowast119901(0) equiv Slowast

119901for a class

of 119901-valent starlike functions in D Denoted by Slowast(119901 120574) thesubclass of Slowast

119901(120574) consists of functions 119891 isin A

119901for which

100381610038161003816100381610038161003816100381610038161003816

1199111198911015840(119911)

119891 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120574 119911 isin D (4)

On the other hand a function 119891 isin A119901is said to be 119901-valently

convex of order 120574 (0 le 120574 lt 119901) in D if 119891 satisfies

Re1 +11991111989110158401015840(119911)

1198911015840 (119911) gt 120574 119911 isin D (5)

We denote this class byK119901(120574)which was introduced by Owa

[11] In particular we set K119901(0) equiv K

119901for a class of 119901-

valent convex functions inD Analogously we denote that thesubclass K(119901 120574) of K

119901(120574) consists of functions 119891 isin A

119901for

which100381610038161003816100381610038161003816100381610038161003816

1 +11991111989110158401015840(119911)

1198911015840 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120574 119911 isin D (6)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 703139 6 pageshttpdxdoiorg1011552014703139

2 Abstract and Applied Analysis

By using the Alexander-type criterion it follows that

119891 (119911) isinK119901(120574) lArrrArr

1199111198911015840(119911)

119901isin Slowast

119901(120574) (7)

The statement is also true if we replace Slowast119901(120574) and K

119901(120574) by

Slowast(119901 120574) andK(119901 120574) respectively Moreover we note that

Slowast(119901 120574) sub S

lowast

119901(120574) sub S

lowast

119901 K (119901 120574) subK

119901(120574) subK

119901

(8)

A function 119891 isin US119901(120575 120574) is said to be 120575-uniformly 119901-

valent starlike of order 120574 (minus1 le 120574 lt 119901 120575 ge 0) inD if119891 satisfies

Re1199111198911015840(119911)

119891 (119911) gt 120575

100381610038161003816100381610038161003816100381610038161003816

1199111198911015840(119911)

119891 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

+ 120574 119911 isin D (9)

Furthermore a function 119891 isin UK119901(120575 120574) is said to be 120575-

uniformly 119901-valent starlike of order 120574 (minus1 le 120574 lt 119901 120575 ge 0)in D if 119891 satisfies

Re1 +11991111989110158401015840(119911)

1198911015840 (119911) gt 120575

100381610038161003816100381610038161003816100381610038161003816

1 +11991111989110158401015840(119911)

1198911015840 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

+ 120574 119911 isin D

(10)

Both US119901(120575 120574) and UK

119901(120575 120574) are comprehensive classes

of analytic functions that include various classes of analyticunivalent functions as well as some very well-known onesFor example in the case 119901 = 1 we have US

1(120575 120574) equiv

US(120575 120574) and UK1(120575 120574) equiv UK(120575 120574) which are introduced

by Bharati et al [12] For 120574 = 0 UK1(120575 0) equiv UK(120575) is

the class of 120575-uniformly convex function [13] In the specialcase 119901 = 120575 = 1 and 120574 = 0 the class US

1(1 0) equiv US

of uniformly starlike functions and UK1(1 0) equiv UK of

uniformly convex functions were introduced by Goodman[14 15] Using the Alexander type relation statement (7)holds for theUK

119901(120573 120574) andUS

119901(120573 120574) that is

119891 (119911) isin UK119901(120573 120574) lArrrArr

1199111198911015840(119911)

119901isin US

119901(120573 120574) (11)

Many researchers have studied the geometric propertiesof integral operators The common investigation is findingsufficient conditions of integral operators in order to trans-form analytic functions into classes with each of those men-tioned properties The well-known integral transformationdefining a subclass of univalent functions was introduced byAlexander in [16] It is of the following form

119865 [119891] (119911) = int

119911

0

119891 (119905)

119905119889119905 (12)

In [17] Kim and Merkes extended the integral operator (12)by introducing a complex parameter 120572 as

119865120572[119891] (119911) = int

119911

0

(119891(119905)

119905)

120572

119889119905 (13)

Another object of investigation for the studies of the integraloperator by Pfaltzgraff [18] is 119866120572 defined by

119866120572[119891] (119911) = int

119911

0

(1198911015840(119905))120572

119889119905 (14)

Until now the various generalized form of the integraloperators 119865120572 in (13) and 119866120572 in (14) has been investigatedHowever Breaz and Stanciu [19] introduced and studied themore general form of integral operator 119869120572120583

119899 A119899 timesA119899 rarr A

which is

119869120572120583

119899[119891 119892] (119911) = int

119911

0

119899

prod

119896=1

(119891119896(119905)

119905)

120572119896

(1198921015840

119896(119905))120583119896119889119905 (15)

By setting appropriate values for the parameters 119899 120572 and120583 integral operators that have been previously introducedcan be obtained In particular if 120583

119896= 0 then the integral

operator 1198691205720119899

becomes the integral operator 119865120572119899introduced by

D Breaz and N Breaz [20] Also when 120572119896= 0 the integral

operator 1198690120583119899

is exactly the integral operator 119866120572119899defined by

Breaz et al [21] The specialized form of 119865120572119899and 119866120572

119899involving

the Bessel functions was introduced and studied in [22ndash24]In addition the specific case 119899 = 1 for 119869120572120583

119899in (15) 119869120572120583

1= 119869120572120583

was investigated by Pescar in [25] The univalence and theirproperties of the integral operators are reported in [26ndash30]

In [31] Bulut developed the integral operator 119868120572120583119899

A119899119901times

A119899119901rarr A

119901which extends the class of analytic functions A

to the class of 119901-valent functionsA119901 that is

119868120572120583

119901119899[119891 119892] (119911) = int

119911

0

119901119905119901minus1

119899

prod

119896=1

(119891119896(119905)

119905119901)

120572119896

(1198921015840

119896(119905)

119901119905119901minus1)

120583119896

119889119905 (16)

By setting 120583119896= 0 and 120572

119896= 0 we obtain the integral

operators 1198681205720119901119899

= 119865120572

119901119899and 1198680120583119901119899 = 119866

120583

119901119899 respectively whichwere introduced by Frasin [32] Also some properties of theseintegral operators have been studied in [32ndash34]

Recently many authors modified integral operators asso-ciated with differential operator such as Salagean operator[35] Ruscheweyh operator [36] Al-Oboudi operator [37]and Carlson-Shaffer operator [38] In [39] Frasin investi-gated one of the generalized integral operators by using theHadamard product to demonstrate most of the previouslydefined integral operators Frasin [39] defined the integraloperator119867

119899 A119899 timesA119899 rarr A by

119867119899[119891 119892] (119911) = int

119911

0

119899

prod

119896=1

((119891119896lowast 119892119896)(119905)

119905)

120572119896

119889119905 (17)

where (119891 lowast 119892)(119911)119911 = 0 119911 isin D and the Hadamard product isdefined by

(119891 lowast 119892) (119911) equiv 119911 +

infin

sum

119895=2

119886119895119887119895119911119895 119911 isin D (18)

where 119891(119911) = 119911 + suminfin119895=2119886119895119911119895 and 119892(119911) = 119911 + suminfin

119895=2119887119895119911119895 It was

reported that for appropriate functions 119892119896isin A the integral

operator 119867119899generalizes many integral operators introduced

and studied by several authors [40ndash44] Moreover the inte-gral operator 119867

119899is generalized integral operators of those

in (12)ndash(15) In a similar idea 119867119899can be extended to more

generalized onA119899119901timesA119899119901toA119901by

119867119901119899[119891 119892] (119911) = int

119911

0

119901119905119901minus1

119899

prod

119896=1

((119891119896lowast 119892119896)(119905)

119905119901)

120572119896

119889119905 (19)

Abstract and Applied Analysis 3

where (119891 lowast 119892)(119911)119911119901 = 0 119891(119911) = 119911119901+ suminfin

119895=1119886119901+119895119911119901+119895 and

119892(119911) = 119911119901+ suminfin

119895=1119887119901+119895119911119901+119895 and the Hadamard product is

defined by

(119891 lowast 119892) (119911) equiv 119911119901+

infin

sum

119895=1

119886119901+119895119887119901+119895119911119901+119895 119911 isin D (20)

Certainly the integral operator 119867119901119899

generalizes many oper-ators when we choose suitable functions 119892

119896isin A119901(see eg

[45ndash47])In fact we can write the Hadamard product in the form

(119891 lowast 119892)(119911) = 119911119901ℎ(119911) where ℎ is analytic in D and ℎ(0) = 1

Indeed

ℎ (119911) = 1 +

infin

sum

119895=1

119886119901+119895119887119901+119895119911119895 119911 isin D (21)

which usually appears in most of integral operators andalways belongs to the classA

0 That is why we are interested

in replacing the term (119891119896lowast 119892119896)(119911)119911

119901 in the integral operator(19) with general function in A

0 Additionally replacing the

term 119911119901minus1 with a function 120601(119911) isin A

119901minus1yields the integral

operator which is still contained inA119901

We now define the following general integral operatorI120572120601 A1198990rarr A

119901 for 119901 ge 1 120601(119911) isin A

119901minus1 and 120572 =

(1205721 1205722 120572

119899) isin R119899+by

I120572

120601[ℎ] (119911) = int

119911

0

119901120601 (119905)

119899

prod

119896=1

(ℎ119896(119905))120572119896119889119905 (22)

where ℎ119896isin A0for all 119896 = 1 2 119899

The main purpose of the paper is to investigate thesufficient conditions on convexity of the integral operatorI120572120601[ℎ] on classes Slowast

119901(120574) Slowast(119901 120574) and US(120575 120574) of analytic

functions Our main results will be applied to reinstate theresults of former researches with related integral operators

2 Main Results

In this section we investigate sufficient conditions for theconvexity of the integral operatorI120572

120601[ℎ] which is defined by

(22) For the convenience we introduce the transformationoperatorT

119901 A119902rarr A

119901+119902by

T119901(119891) (119911) = 119911

119901119891 (119911) 119911 isin D (23)

where 119891 isin A119902and 119902 is a nonnegative integer In particular

we setT1= T

We now prove a general property which guarantees theconvexity of the proposed integral operator on the classSlowast119901(120574)

Theorem 1 Let 0 le 120582 120574119896lt 119901T(120601) isin Slowast

119901(120582) andT

119901(ℎ119896) isin

Slowast119901(120574119896) for 119896 = 1 2 119899 If 120572 = (120572

1 1205722 120572

119899) isin R119899+satisfies

119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (24)

then the integral operator I120572120601[ℎ] defined by (22) is 119901-valently

convex of order 120582 minus sum119899119896=1120572119896(119901 minus 120574

119896)

Proof From the definition of integral opeartor in (22) weobserve that I120572

120601[ℎ] isin A

119901 By calculating the first derivative

ofI120572120601[ℎ] we obtain

(I120572

120601[ℎ])1015840

(119911) = 119901120601 (119911)

119899

prod

119896=1

(ℎ119896(119911))120572119896 (25)

Differentiating on both sides of (25) logarithmically andmultiplying by 119911 give

119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

=1199111206011015840(119911)

120601 (119911)+

119899

sum

119896=1

120572119896

119911ℎ1015840

119896(119911)

ℎ119896(119911)

(26)

SinceT(120601) isin Slowast119901(120582) it follows that

Re119911T(120601 (119911))

1015840

(119911)

T (120601) (119911) = Re

119911(119911120601 (119911))1015840

(119911)

119911120601 (119911)

= Re1 +1199111206011015840(119911)

120601 (119911) gt 120582

(27)

Also sinceT119901(ℎ119896) isin Slowast119901(120574119896) we have

Re

119911(T119901(ℎ119896))1015840

(119911)

T119901(ℎ119896) (119911)

= Re119911(119911119901ℎ119896)1015840

(119911)

119911119901ℎ119896(119911)

= Re119901 +119911ℎ1015840

119896(119911)

ℎ119896(119911)

gt 120574119896

(28)

From (27) and (28) by taking the real part of (26) we obtain

Re

1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

= Re1 +1199111206011015840(119911)

120601 (119911) +

119899

sum

119896=1

120572119896Re

119911ℎ1015840

119896(119911)

ℎ119896(119911)

gt 120582 +

119899

sum

119896=1

120572119896Re119901 +

119911ℎ1015840

119896(119911)

ℎ119896(119911)

minus 119901

119899

sum

119896=1

120572119896

gt 120582 minus

119899

sum

119896=1

120572119896(119901 minus 120574

119896)

(29)

ThereforeI120572120601[ℎ] is119901-valently convex of order120582minussum119899

119896=1120572119896(119901minus

120574119896)

We note that by suitable functions 120601 isin A119901minus1

ℎ119896isin A0

and (1205721 1205722 120572

119899) isin R119899+in Theorem 1 we obtain the earlier

result For example if 119901 = 1 and 120601(119911) = 1 we obtainTheorem 21 in [44] andTheorems 21 and 22 in [48] If119901 ge 1and 120601(119911) = 119911

119901minus1 by using the Alexander type relation (7)Theorem 1 in [19] is obtained


Using the same method and technique as that inTheorem 1 with the nonnegativity of modulus of complexnumbers we are led easily toTheorem 2Theproof is omitted

Theorem 2 Let T(120601) isin US119901(120575 120582) and T

119901(ℎ119896) isin

US119901(120575119896 120574119896) for 119896 = 1 2 119899 If 120572 = (120572

1 1205722 120572

119899) isin R119899

+

satisfies119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (30)



119896)

Theorem 2 generalizes many results proposed by severalauthors For 119901 = 1 and 120601(119911) = 1 we obtain Theorem 21 in[26] and Theorem 2 in [49] For 119901 ge 1 and 120601(119911) = 119911119901minus1 withthe Alexander type relation (11) we obtainTheorem 2 in [46]andTheorem 21 in [31] AlsoTheorems 21 and 31 in [33] areobtained

The following is a result on the transformation propertyofI120572120601[ℎ] on the class Slowast(119901 120574)

Theorem 3 Let T(120601) isin Slowast(119901 120582) and T119901(ℎ119896) isin Slowast(119901 120574

119896)

for 119896 = 1 2 119899 If 120572 = (1205721 1205722 120572


119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (31)

then the integral operatorI120572120601[ℎ] defined by (22) is in the class

K(119901 120582 minus sum119899

119896=1120572119896(119901 minus 120574

119896)) FurthermoreI120572

120601[ℎ] is 119901-valently


119896)

Proof From (26) we obtain100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

1 +1199111206011015840(119911)

120601 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

+

119899

sum

119896=1

120572119896

100381610038161003816100381610038161003816100381610038161003816

119911ℎ1015840

119896(119911)

ℎ119896(119911)

100381610038161003816100381610038161003816100381610038161003816

(32)

SinceT(120601) isin Slowast(119901 120582) it follows that1003816100381610038161003816100381610038161003816100381610038161003816

119911(T (120601))1015840

(119911)

T (120601) (119911)minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119911120601)1015840

(119911)

119911120601 (119911)minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1 +1199111206011015840(119911)

120601 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120582

(33)

Also sinceT119901(ℎ119896) isin Slowast(119901 120574

119896) we have

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119911(T119901(ℎ119896))1015840

(119911)

T119901(ℎ119896) (119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119911119901ℎ119896)1015840

(119911)

119911119901ℎ119896 (119911)

minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119911ℎ1015840

119896(119911)

ℎ119896(119911)

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120574119896

(34)

By substitution of (33) and (34) into (32) we get100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119901 minus 120582 +

119899

sum

119896=1

120572119896(119901 minus 120574

119896) (35)

ThereforeI120572120601[ℎ] isinK(119901 120574 minussum

119899

119896=1120572119896(119901 minus120573

119896)) It follows that

I120572120601[ℎ] is 119901-valently convex of order 120582 minus sum119899

119896=1120572119896(119901 minus 120574

119896)

Remark 4 The parameter 120572 = (1205721 1205722 120572

119899) in Theorem 3

can be extended to the complex number and assumption (31)becomes

119899

sum

119896=1

10038161003816100381610038161205721198961003816100381610038161003816 (119901 minus 120574119896) le 120582 (36)

That is Theorem 3 can be applied to the integral operatorI120572120601[ℎ] in case 120572 = (120572

1 1205722 120572

119899) isin C119899

Over the past few decades there are many studies onthe sufficient conditions that make the integral operatorsunivalent In fact the class of convex functions is a subclass ofthe class of all univalent functions inD Thus it is interestingto observe that many results on the univalence property ofintegral operators follow the convexity property according tomain results especially Theorem 3 or Remark 4

We now consider the integral operator119867119899 A119899 timesA119899 rarr

A defined in (17) In order to obtain the convexity of theintegral operator119867

119899byTheorem 3orRemark 4we set120601(119911) =

1 119901 = 1 120574119896= 0 120572 = (120572

1 1205722 120572

119899) isin C119899 and

ℎ119896(119911) =

(119891119896lowast 119892119896) (119911)

119911isin A0 119896 = 1 2 119899 (37)

where 119891119896 119892119896isin A Other than that the univalent property

of 119867119899is also obtained This implies Theorem 31 of Frasin

in [39] Moreover Frasin [39] noticed that for suitablefunctions 119892

119896isin A the integral operator 119867

119899generalizes

many operators introduced by several authors for instanceTheorem 1 in [20] Theorem 21 in [41] Theorem 23 in [42]and Theorem 23 in [43] It is noteworthy to say that undersame assumptions the former researches obtain only theunivalence while we obtain the stronger result which is theconvexity

Our results can be used to explain the convexity of theother integral operators that are related to the Hadamadproduct as described next

Remark 5 For 119901 ge 1 we set 120601(119911) = 119911119901minus1 and take

ℎ119896(119911) =

(119891119896lowast 119892119896) (119911)

119911119901isin A0 119896 = 1 2 119899 (38)

where 119891119896 119892119896isin A119901 Then we can apply the main results to

discuss the convexity of the integral operator119867119901119899

defined by(19)

Remark 6 Themain results are also applicable to the integraloperator 119866

119901119899 A119899119901timesA119899119901rarr A

119901of the following form

119866119901119899[119891 119892] (119911) = int

119911

0

119901119905119901minus1

119899

prod

119896=1

((119891119896lowast 119892119896)1015840(119905)

119901119905119901minus1)

120572119896

119889119905 (39)


where 119891119896 119892119896isin A119901 In order to apply the main results by the

Alexander-type criterion we note that

T119901((119891119896lowast 119892119896)1015840

(119911)

119901119911119901minus1) isin S

lowast

119901(120574)

lArrrArr119911(119891119896lowast 119892119896)1015840

(119911)

119901isin Slowast

119901(120574)

lArrrArr (119891119896lowast 119892119896) (119911) isinK

119901(120574)

(40)

The above statement also holds for the pairs of classesUS119901(120573 120574) minusUK

119901(120573 120574) and S(119901 120574) minusK(119901 120574)

Remark 7 For suitable functions 119892119896isin A119901 by Remarks 5 and

6 we obtain new results for the convexity of other integraloperators for example F119899119898

119901120575119897and G119899119898

119901120575119897in [45] F

119901119899119897120575and

G119901119899119897120575

in [46]F119901119898119897120583

andG119901119898119897120583

in [50] and so forth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors sincerely thank the reviewers for their valu-able suggestions and useful comments that improved thepresentation of the paper This research was supported byScience Achievement Scholarship of Thailand (SAST) andDepartment of Mathematics Faculty of Science Chiang MaiUniversity

References

[1] C Pommerenke Univalent Functions Vandemhoeck andRuprecht Gottingen Germany 1975

[2] A W GoodmanUnivalent Functions vol 1-2 Mariner TampaFla USA 1983

[3] M I S Robertson ldquoOn the theory of univalent functionsrdquoAnnals of Mathematics vol 37 no 2 pp 374ndash408 1936

[4] A Schild ldquoOn a class of univalent star shaped mappingsrdquoProceedings of the American Mathematical Society vol 9 pp751ndash757 1958

[5] E P Merkes ldquoOn products of starlike functionsrdquo Proceedings ofthe American Mathematical Society vol 13 pp 479ndash492 2013

[6] MNunokawa ldquoOn themultivalent functionsrdquo Indian Journal ofPure and Applied Mathematics vol 20 no 6 pp 577ndash582 1989

[7] M Nunokawa and S Owa ldquoOn certain subclass of analyticfunctionsrdquo Indian Journal of Pure and AppliedMathematics vol19 no 1 pp 51ndash54 1988

[8] J Dziok ldquoApplications of the Jack lemmardquo Acta MathematicaHungarica vol 105 no 1-2 pp 93ndash102 2004

[9] A E Livingston ldquop-valent close-to-convex functionsrdquo SoochowJournal of Mathematics vol 21 pp 161ndash179 1995

[10] D A Patil and N K Thakare ldquoOn convex hulls and extremepoints of p-valent starlike and convex classwswith applicationsrdquoBulletin Mathematique de la Societe des Sciences Mathematiquesde la Republique Socialiste de Roumanie vol 27 no 75 pp 145ndash160 1983

[11] S Owa ldquoOn certain classes of 119901-valent functions with negativecoefficientsrdquo Simon Stevin A Quarterly Journal of Pure andApplied Mathematics vol 59 no 4 pp 385ndash402 1985

[12] R Bharati R Parvatham and A Swaminathan ldquoOn subclassesof uniformly convex functions and corresponding class ofstarlike functionsrdquoTamkang Journal ofMathematics vol 28 no1 pp 17ndash32 1997

[13] S Kanas and A Wisniowska ldquoConic regions and 119896-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999

[14] A W Goodman ldquoOn uniformly starlike functionsrdquo Journal ofMathematical Analysis and Applications vol 155 pp 364ndash3701991

[15] A W Goodman ldquoOn uniformly convex functionsrdquo PolskaAkademia Nauk Annales Polonici Mathematici vol 56 no 1pp 87ndash92 1991

[16] J W Alexander ldquoFunctions which map the interior of the circleupon simple regionsrdquo Annals of Mathematics vol 14 pp 12ndash221915

[17] Y J Kim and E P Merkes ldquoOn an integral of powers of aspirallike functionrdquo Kyungpook Mathematical Journal vol 12pp 249ndash252 1972

[18] J A Pfaltzgraff ldquoUnivalence of the integral of f 1015840(z)120582rdquo TheBulletin of the London Mathematical Society vol 7 no 3 pp254ndash256 1975

[19] D Breaz and L Stanciu ldquoSome properties of a general integraloperatorrdquo Bulletin of the Transilvania University of Brasov SeriesIII Mathematics Informatics Physics vol 5 no 54 pp 67ndash722012

[20] D Breaz and N Breaz ldquoTwo integral operatorrdquo Studia Univer-sitatis Babes-Bolyai vol 47 no 3 pp 13ndash19 2002

[21] D Breaz S Owa and N Breaz ldquoA new integral univalent oper-atorrdquoActa Universitatis Apulensis Mathematics-Informatics no16 pp 11ndash16 2008

[22] E Deniz H Orhan andHM Srivastava ldquoSome sufficient con-ditions for univalence of certain families of integral operatorsinvolving generalized Bessel functionsrdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 883ndash917 2011

[23] E Deniz ldquoConvexity of integral operators involving generalizedBessel functionsrdquo Integral Transforms and Special Functions vol24 no 3 pp 201ndash216 2013

[24] B A Frasin ldquoSufficient conditions for integral operator definedby Bessel functionsrdquo Journal of Mathematical Inequalities vol4 no 2 pp 301ndash306 2010

[25] V Pescar ldquoThe univalence and the convexity properties for anew integral operatorrdquo StudiaUniversitatis Babes-Bolyai vol 56no 4 pp 65ndash69 2011

[26] D Breaz ldquoA convexity property for an integral operator on theclass S

119901(120573)rdquo Journal of Inequalities and Applications vol 2008

Article ID 143869 4 pages 2008[27] B A Frasin ldquoSome sufficient conditions for certain integral

operatorrdquo Journal of Inequalities and Applications vol 2 no 4pp 527ndash535 2008

[28] S Bulut ldquoA note on the paper of Breaz and Guneyrdquo Journal ofMathematical Inequalities vol 2 no 4 pp 549ndash553 2008

[29] D Breaz andV Pescar ldquoUnivalence conditions for some generalintegral operatorsrdquo Banach Journal of Mathematical Analysisvol 2 no 1 pp 53ndash58 2008

[30] D Breaz and H O Guney ldquoThe integral operator on the classesSlowast120572(b) and C

120572(b)rdquo Journal of Mathematical Inequalities vol 2

no 1 pp 97ndash100 2008


[31] S Bulut ldquoConvexity properties of a new general integral oper-ator of p-valent functionsrdquo Mathematical Journal of OkayamaUniversity vol 56 pp 171ndash178 2014

[32] B A Frasin ldquoNew general integral operators of 119901-valent func-tionsrdquo Journal of Inequalities in Pure and Applied Mathematicsvol 10 no 4 article 109 2009

[33] B A Frasin ldquoConvexity of integral operators of 119901-valentfunctionsrdquo Mathematical and Computer Modelling vol 51 no5-6 pp 601ndash605 2010

[34] A Mohammed M Darus and D Breaz ldquoNew criterion forstarlike integral operatorsrdquo Analysis inTheory and Applicationsvol 29 no 1 pp 21ndash26 2013

[35] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372Springer Berlin Germany 1983

[36] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the AmericanMathematical Society vol 49 no 1 pp 109ndash115 1975

[37] F M Al-Oboudi ldquoOn univalent functions defined by a gener-alized Salagean operatorrdquo International Journal of Mathematicsand Mathematical Sciences vol 2004 no 27 pp 1429ndash14362004

[38] B C Carlson and D B Shaffer ldquoStarlike and prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 1984

[39] B A Frasin ldquoGeneral integral operator defined by HadamardproductrdquoMatematichki Vesnik vol 62 no 2 pp 127ndash136 2010

[40] D Breaz H Guney and G Salagean ldquoA new integral operatorrdquoin Proceedings of the 7th Joint Conference on Mathematics andComputer Science Cluj Romania July 2008

[41] C Selvaraj and K R Karthikeyan ldquoSufficient conditions forunivalence of a general integral operatorrdquo Bulletin of the KoreanMathematical Society vol 46 no 2 pp 367ndash372 2009

[42] G I Oros and D Breaz ldquoSufficient conditions for univalenceof an integral operatorrdquo Journal of Inequalities and Applicationsvol 2008 Article ID 127645 7 pages 2008

[43] S Bulut ldquoSufficient conditions for univalence of an integraloperator defined by Al-Oboudi differential operatorrdquo Journalof Inequalities and Applications vol 2008 Article ID 9570425 pages 2008

[44] S Bulut ldquoSome properties for an integral operator defined byAl-Oboudi differential operatorrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 4 article 115 2008

[45] R M El-Ashwah M K Aouf A Shamandy and S M El-DeebldquoProperties of certain classes of p-valent functions associatedwith new integral operatorsrdquoThe American Journal of Pure andApplied Mathematics vol 2 pp 79ndash80 2013

[46] G Saltik Ayhanoz and E Kadioglu ldquoSome results of p-valentfunctions defined by integral operatorsrdquo Acta UniversitatisApulensis Mathematics Informatics no 32 pp 69ndash85 2012

[47] G Saltik E Deniz and E Kadioglu ldquoTwo new general 119901-valentintegral operatorsrdquoMathematical and Computer Modelling vol52 no 9-10 pp 1605ndash1609 2010

[48] D Breaz and N Breaz ldquoSome convexity properties for a generalintegral operatorrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 no 5 article 177 8 pages 2006

[49] A Mohammed M Darus and D Breaz ldquoSome propertiesfor certain integral operatorsrdquo Acta Universitatis ApulensismdashMatematicsmdashInformatics no 23 pp 79ndash89 2010

[50] E Deniz M Caglar and H Orhan ldquoSome convexity propertiesfor two new 119901-valent integral operatorsrdquo Hacettepe Journal ofMathematics and Statistics vol 40 no 6 pp 829ndash837 2011

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


By using the Alexander-type criterion it follows that

119891 (119911) isinK119901(120574) lArrrArr

1199111198911015840(119911)

119901isin Slowast

119901(120574) (7)

The statement is also true if we replace Slowast119901(120574) and K

119901(120574) by

Slowast(119901 120574) andK(119901 120574) respectively Moreover we note that

Slowast(119901 120574) sub S

lowast

119901(120574) sub S

lowast

119901 K (119901 120574) subK

119901(120574) subK

119901

(8)

A function 119891 isin US119901(120575 120574) is said to be 120575-uniformly 119901-

valent starlike of order 120574 (minus1 le 120574 lt 119901 120575 ge 0) inD if119891 satisfies

Re1199111198911015840(119911)

119891 (119911) gt 120575

100381610038161003816100381610038161003816100381610038161003816

1199111198911015840(119911)

119891 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

+ 120574 119911 isin D (9)

Furthermore a function 119891 isin UK119901(120575 120574) is said to be 120575-

uniformly 119901-valent starlike of order 120574 (minus1 le 120574 lt 119901 120575 ge 0)in D if 119891 satisfies

Re1 +11991111989110158401015840(119911)

1198911015840 (119911) gt 120575

100381610038161003816100381610038161003816100381610038161003816

1 +11991111989110158401015840(119911)

1198911015840 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

+ 120574 119911 isin D

(10)

Both US119901(120575 120574) and UK

119901(120575 120574) are comprehensive classes

of analytic functions that include various classes of analyticunivalent functions as well as some very well-known onesFor example in the case 119901 = 1 we have US

1(120575 120574) equiv

US(120575 120574) and UK1(120575 120574) equiv UK(120575 120574) which are introduced

by Bharati et al [12] For 120574 = 0 UK1(120575 0) equiv UK(120575) is

the class of 120575-uniformly convex function [13] In the specialcase 119901 = 120575 = 1 and 120574 = 0 the class US

1(1 0) equiv US

of uniformly starlike functions and UK1(1 0) equiv UK of

uniformly convex functions were introduced by Goodman[14 15] Using the Alexander type relation statement (7)holds for theUK

119901(120573 120574) andUS

119901(120573 120574) that is

119891 (119911) isin UK119901(120573 120574) lArrrArr

1199111198911015840(119911)

119901isin US

119901(120573 120574) (11)

Many researchers have studied the geometric propertiesof integral operators The common investigation is findingsufficient conditions of integral operators in order to trans-form analytic functions into classes with each of those men-tioned properties The well-known integral transformationdefining a subclass of univalent functions was introduced byAlexander in [16] It is of the following form

119865 [119891] (119911) = int

119911

0

119891 (119905)

119905119889119905 (12)

In [17] Kim and Merkes extended the integral operator (12)by introducing a complex parameter 120572 as

119865120572[119891] (119911) = int

119911

0

(119891(119905)

119905)

120572

119889119905 (13)

Another object of investigation for the studies of the integraloperator by Pfaltzgraff [18] is 119866120572 defined by

119866120572[119891] (119911) = int

119911

0

(1198911015840(119905))120572

119889119905 (14)

Until now the various generalized form of the integraloperators 119865120572 in (13) and 119866120572 in (14) has been investigatedHowever Breaz and Stanciu [19] introduced and studied themore general form of integral operator 119869120572120583

119899 A119899 timesA119899 rarr A

which is

119869120572120583

119899[119891 119892] (119911) = int

119911

0

119899

prod

119896=1

(119891119896(119905)

119905)

120572119896

(1198921015840

119896(119905))120583119896119889119905 (15)

By setting appropriate values for the parameters 119899 120572 and120583 integral operators that have been previously introducedcan be obtained In particular if 120583

119896= 0 then the integral

operator 1198691205720119899

becomes the integral operator 119865120572119899introduced by

D Breaz and N Breaz [20] Also when 120572119896= 0 the integral

operator 1198690120583119899

is exactly the integral operator 119866120572119899defined by

Breaz et al [21] The specialized form of 119865120572119899and 119866120572

119899involving

the Bessel functions was introduced and studied in [22ndash24]In addition the specific case 119899 = 1 for 119869120572120583

119899in (15) 119869120572120583

1= 119869120572120583

was investigated by Pescar in [25] The univalence and theirproperties of the integral operators are reported in [26ndash30]

In [31] Bulut developed the integral operator 119868120572120583119899

A119899119901times

A119899119901rarr A

119901which extends the class of analytic functions A

to the class of 119901-valent functionsA119901 that is

119868120572120583

119901119899[119891 119892] (119911) = int

119911

0

119901119905119901minus1

119899

prod

119896=1

(119891119896(119905)

119905119901)

120572119896

(1198921015840

119896(119905)

119901119905119901minus1)

120583119896

119889119905 (16)

By setting 120583119896= 0 and 120572

119896= 0 we obtain the integral

operators 1198681205720119901119899

= 119865120572

119901119899and 1198680120583119901119899 = 119866

120583

119901119899 respectively whichwere introduced by Frasin [32] Also some properties of theseintegral operators have been studied in [32ndash34]

Recently many authors modified integral operators asso-ciated with differential operator such as Salagean operator[35] Ruscheweyh operator [36] Al-Oboudi operator [37]and Carlson-Shaffer operator [38] In [39] Frasin investi-gated one of the generalized integral operators by using theHadamard product to demonstrate most of the previouslydefined integral operators Frasin [39] defined the integraloperator119867

119899 A119899 timesA119899 rarr A by

119867119899[119891 119892] (119911) = int

119911

0

119899

prod

119896=1

((119891119896lowast 119892119896)(119905)

119905)

120572119896

119889119905 (17)

where (119891 lowast 119892)(119911)119911 = 0 119911 isin D and the Hadamard product isdefined by

(119891 lowast 119892) (119911) equiv 119911 +

infin

sum

119895=2

119886119895119887119895119911119895 119911 isin D (18)

where 119891(119911) = 119911 + suminfin119895=2119886119895119911119895 and 119892(119911) = 119911 + suminfin

119895=2119887119895119911119895 It was

reported that for appropriate functions 119892119896isin A the integral

operator 119867119899generalizes many integral operators introduced

and studied by several authors [40ndash44] Moreover the inte-gral operator 119867

119899is generalized integral operators of those

in (12)ndash(15) In a similar idea 119867119899can be extended to more

generalized onA119899119901timesA119899119901toA119901by

119867119901119899[119891 119892] (119911) = int

119911

0

119901119905119901minus1

119899

prod

119896=1

((119891119896lowast 119892119896)(119905)

119905119901)

120572119896

119889119905 (19)



119895=1119886119901+119895119911119901+119895 and

119892(119911) = 119911119901+ suminfin


defined by

(119891 lowast 119892) (119911) equiv 119911119901+

infin

sum

119895=1

119886119901+119895119887119901+119895119911119901+119895 119911 isin D (20)






Indeed

ℎ (119911) = 1 +

infin

sum

119895=1

119886119901+119895119887119901+119895119911119895 119911 isin D (21)










119901 for 119901 ge 1 120601(119911) isin A

119901minus1 and 120572 =

(1205721 1205722 120572

119899) isin R119899+by

I120572

120601[ℎ] (119911) = int

119911

0

119901120601 (119905)

119899

prod

119896=1

(ℎ119896(119905))120572119896119889119905 (22)





2 Main Results




119901 A119902rarr A

119901+119902by

T119901(119891) (119911) = 119911

119901119891 (119911) 119911 isin D (23)


we setT1= T



119901(120582) andT

119901(ℎ119896) isin

Slowast119901(120574119896) for 119896 = 1 2 119899 If 120572 = (120572

1 1205722 120572


119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (24)



119896)


120601[ℎ] isin A



(I120572

120601[ℎ])1015840

(119911) = 119901120601 (119911)

119899

prod

119896=1

(ℎ119896(119911))120572119896 (25)


119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

=1199111206011015840(119911)

120601 (119911)+

119899

sum

119896=1

120572119896

119911ℎ1015840

119896(119911)

ℎ119896(119911)

(26)


Re119911T(120601 (119911))

1015840

(119911)

T (120601) (119911) = Re

119911(119911120601 (119911))1015840

(119911)

119911120601 (119911)

= Re1 +1199111206011015840(119911)

120601 (119911) gt 120582

(27)


Re

119911(T119901(ℎ119896))1015840

(119911)

T119901(ℎ119896) (119911)

= Re119911(119911119901ℎ119896)1015840

(119911)

119911119901ℎ119896(119911)

= Re119901 +119911ℎ1015840

119896(119911)

ℎ119896(119911)

gt 120574119896

(28)


Re

1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

= Re1 +1199111206011015840(119911)

120601 (119911) +

119899

sum

119896=1

120572119896Re

119911ℎ1015840

119896(119911)

ℎ119896(119911)

gt 120582 +

119899

sum

119896=1

120572119896Re119901 +

119911ℎ1015840

119896(119911)

ℎ119896(119911)

minus 119901

119899

sum

119896=1

120572119896

gt 120582 minus

119899

sum

119896=1

120572119896(119901 minus 120574

119896)

(29)


119896=1120572119896(119901minus

120574119896)


ℎ119896isin A0

and (1205721 1205722 120572







119901(ℎ119896) isin

US119901(120575119896 120574119896) for 119896 = 1 2 119899 If 120572 = (120572

1 1205722 120572

119899) isin R119899

+

satisfies119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (30)



119896)




119896)

for 119896 = 1 2 119899 If 120572 = (1205721 1205722 120572


119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (31)


K(119901 120582 minus sum119899

119896=1120572119896(119901 minus 120574


120601[ℎ] is 119901-valently


119896)


1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

1 +1199111206011015840(119911)

120601 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

+

119899

sum

119896=1

120572119896

100381610038161003816100381610038161003816100381610038161003816

119911ℎ1015840

119896(119911)

ℎ119896(119911)

100381610038161003816100381610038161003816100381610038161003816

(32)


119911(T (120601))1015840

(119911)

T (120601) (119911)minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119911120601)1015840

(119911)

119911120601 (119911)minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1 +1199111206011015840(119911)

120601 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120582

(33)


119896) we have

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119911(T119901(ℎ119896))1015840

(119911)

T119901(ℎ119896) (119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119911119901ℎ119896)1015840

(119911)

119911119901ℎ119896 (119911)

minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119911ℎ1015840

119896(119911)

ℎ119896(119911)

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120574119896

(34)


1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119901 minus 120582 +

119899

sum

119896=1

120572119896(119901 minus 120574

119896) (35)


119899

119896=1120572119896(119901 minus120573



119896=1120572119896(119901 minus 120574

119896)




119899

sum

119896=1

10038161003816100381610038161205721198961003816100381610038161003816 (119901 minus 120574119896) le 120582 (36)


1 1205722 120572

119899) isin C119899





1 119901 = 1 120574119896= 0 120572 = (120572

1 1205722 120572

119899) isin C119899 and

ℎ119896(119911) =

(119891119896lowast 119892119896) (119911)

119911isin A0 119896 = 1 2 119899 (37)





119899generalizes




ℎ119896(119911) =

(119891119896lowast 119892119896) (119911)

119911119901isin A0 119896 = 1 2 119899 (38)



defined by(19)


119901119899 A119899119901timesA119899119901rarr A


119866119901119899[119891 119892] (119911) = int

119911

0

119901119905119901minus1

119899

prod

119896=1

((119891119896lowast 119892119896)1015840(119905)

119901119905119901minus1)

120572119896

119889119905 (39)




T119901((119891119896lowast 119892119896)1015840

(119911)

119901119911119901minus1) isin S

lowast

119901(120574)

lArrrArr119911(119891119896lowast 119892119896)1015840

(119911)

119901isin Slowast

119901(120574)


119901(120574)

(40)


119901(120573 120574) and S(119901 120574) minusK(119901 120574)



119901120575119897and G119899119898

119901120575119897in [45] F

119901119899119897120575and

G119901119899119897120575

in [46]F119901119898119897120583

andG119901119898119897120583




Acknowledgments


References


































no 1 pp 97ndash100 2008





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895=1119886119901+119895119911119901+119895 and

119892(119911) = 119911119901+ suminfin


defined by

(119891 lowast 119892) (119911) equiv 119911119901+

infin

sum

119895=1

119886119901+119895119887119901+119895119911119901+119895 119911 isin D (20)






Indeed

ℎ (119911) = 1 +

infin

sum

119895=1

119886119901+119895119887119901+119895119911119895 119911 isin D (21)










119901 for 119901 ge 1 120601(119911) isin A

119901minus1 and 120572 =

(1205721 1205722 120572

119899) isin R119899+by

I120572

120601[ℎ] (119911) = int

119911

0

119901120601 (119905)

119899

prod

119896=1

(ℎ119896(119905))120572119896119889119905 (22)





2 Main Results




119901 A119902rarr A

119901+119902by

T119901(119891) (119911) = 119911

119901119891 (119911) 119911 isin D (23)


we setT1= T



119901(120582) andT

119901(ℎ119896) isin

Slowast119901(120574119896) for 119896 = 1 2 119899 If 120572 = (120572

1 1205722 120572


119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (24)



119896)


120601[ℎ] isin A



(I120572

120601[ℎ])1015840

(119911) = 119901120601 (119911)

119899

prod

119896=1

(ℎ119896(119911))120572119896 (25)


119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

=1199111206011015840(119911)

120601 (119911)+

119899

sum

119896=1

120572119896

119911ℎ1015840

119896(119911)

ℎ119896(119911)

(26)


Re119911T(120601 (119911))

1015840

(119911)

T (120601) (119911) = Re

119911(119911120601 (119911))1015840

(119911)

119911120601 (119911)

= Re1 +1199111206011015840(119911)

120601 (119911) gt 120582

(27)


Re

119911(T119901(ℎ119896))1015840

(119911)

T119901(ℎ119896) (119911)

= Re119911(119911119901ℎ119896)1015840

(119911)

119911119901ℎ119896(119911)

= Re119901 +119911ℎ1015840

119896(119911)

ℎ119896(119911)

gt 120574119896

(28)


Re

1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

= Re1 +1199111206011015840(119911)

120601 (119911) +

119899

sum

119896=1

120572119896Re

119911ℎ1015840

119896(119911)

ℎ119896(119911)

gt 120582 +

119899

sum

119896=1

120572119896Re119901 +

119911ℎ1015840

119896(119911)

ℎ119896(119911)

minus 119901

119899

sum

119896=1

120572119896

gt 120582 minus

119899

sum

119896=1

120572119896(119901 minus 120574

119896)

(29)


119896=1120572119896(119901minus

120574119896)


ℎ119896isin A0

and (1205721 1205722 120572







119901(ℎ119896) isin

US119901(120575119896 120574119896) for 119896 = 1 2 119899 If 120572 = (120572

1 1205722 120572

119899) isin R119899

+

satisfies119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (30)



119896)




119896)

for 119896 = 1 2 119899 If 120572 = (1205721 1205722 120572


119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (31)


K(119901 120582 minus sum119899

119896=1120572119896(119901 minus 120574


120601[ℎ] is 119901-valently


119896)


1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

1 +1199111206011015840(119911)

120601 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

+

119899

sum

119896=1

120572119896

100381610038161003816100381610038161003816100381610038161003816

119911ℎ1015840

119896(119911)

ℎ119896(119911)

100381610038161003816100381610038161003816100381610038161003816

(32)


119911(T (120601))1015840

(119911)

T (120601) (119911)minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119911120601)1015840

(119911)

119911120601 (119911)minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1 +1199111206011015840(119911)

120601 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120582

(33)


119896) we have

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119911(T119901(ℎ119896))1015840

(119911)

T119901(ℎ119896) (119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119911119901ℎ119896)1015840

(119911)

119911119901ℎ119896 (119911)

minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119911ℎ1015840

119896(119911)

ℎ119896(119911)

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120574119896

(34)


1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119901 minus 120582 +

119899

sum

119896=1

120572119896(119901 minus 120574

119896) (35)


119899

119896=1120572119896(119901 minus120573



119896=1120572119896(119901 minus 120574

119896)




119899

sum

119896=1

10038161003816100381610038161205721198961003816100381610038161003816 (119901 minus 120574119896) le 120582 (36)


1 1205722 120572

119899) isin C119899





1 119901 = 1 120574119896= 0 120572 = (120572

1 1205722 120572

119899) isin C119899 and

ℎ119896(119911) =

(119891119896lowast 119892119896) (119911)

119911isin A0 119896 = 1 2 119899 (37)





119899generalizes




ℎ119896(119911) =

(119891119896lowast 119892119896) (119911)

119911119901isin A0 119896 = 1 2 119899 (38)



defined by(19)


119901119899 A119899119901timesA119899119901rarr A


119866119901119899[119891 119892] (119911) = int

119911

0

119901119905119901minus1

119899

prod

119896=1

((119891119896lowast 119892119896)1015840(119905)

119901119905119901minus1)

120572119896

119889119905 (39)




T119901((119891119896lowast 119892119896)1015840

(119911)

119901119911119901minus1) isin S

lowast

119901(120574)

lArrrArr119911(119891119896lowast 119892119896)1015840

(119911)

119901isin Slowast

119901(120574)


119901(120574)

(40)


119901(120573 120574) and S(119901 120574) minusK(119901 120574)



119901120575119897and G119899119898

119901120575119897in [45] F

119901119899119897120575and

G119901119899119897120575

in [46]F119901119898119897120583

andG119901119898119897120583




Acknowledgments


References


































no 1 pp 97ndash100 2008





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901(ℎ119896) isin

US119901(120575119896 120574119896) for 119896 = 1 2 119899 If 120572 = (120572

1 1205722 120572

119899) isin R119899

+

satisfies119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (30)



119896)




119896)

for 119896 = 1 2 119899 If 120572 = (1205721 1205722 120572


119899

sum

119896=1

120572119896(119901 minus 120574

119896) le 120582 (31)


K(119901 120582 minus sum119899

119896=1120572119896(119901 minus 120574


120601[ℎ] is 119901-valently


119896)


1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

1 +1199111206011015840(119911)

120601 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

+

119899

sum

119896=1

120572119896

100381610038161003816100381610038161003816100381610038161003816

119911ℎ1015840

119896(119911)

ℎ119896(119911)

100381610038161003816100381610038161003816100381610038161003816

(32)


119911(T (120601))1015840

(119911)

T (120601) (119911)minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119911120601)1015840

(119911)

119911120601 (119911)minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1 +1199111206011015840(119911)

120601 (119911)minus 119901

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120582

(33)


119896) we have

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119911(T119901(ℎ119896))1015840

(119911)

T119901(ℎ119896) (119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119911(119911119901ℎ119896)1015840

(119911)

119911119901ℎ119896 (119911)

minus 119901

1003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119911ℎ1015840

119896(119911)

ℎ119896(119911)

100381610038161003816100381610038161003816100381610038161003816

lt 119901 minus 120574119896

(34)


1 +119911(I120572120601[ℎ])10158401015840

(119911)

(I120572120601[ℎ])1015840

(119911)

minus 119901

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119901 minus 120582 +

119899

sum

119896=1

120572119896(119901 minus 120574

119896) (35)


119899

119896=1120572119896(119901 minus120573



119896=1120572119896(119901 minus 120574

119896)




119899

sum

119896=1

10038161003816100381610038161205721198961003816100381610038161003816 (119901 minus 120574119896) le 120582 (36)


1 1205722 120572

119899) isin C119899





1 119901 = 1 120574119896= 0 120572 = (120572

1 1205722 120572

119899) isin C119899 and

ℎ119896(119911) =

(119891119896lowast 119892119896) (119911)

119911isin A0 119896 = 1 2 119899 (37)





119899generalizes




ℎ119896(119911) =

(119891119896lowast 119892119896) (119911)

119911119901isin A0 119896 = 1 2 119899 (38)



defined by(19)


119901119899 A119899119901timesA119899119901rarr A


119866119901119899[119891 119892] (119911) = int

119911

0

119901119905119901minus1

119899

prod

119896=1

((119891119896lowast 119892119896)1015840(119905)

119901119905119901minus1)

120572119896

119889119905 (39)




T119901((119891119896lowast 119892119896)1015840

(119911)

119901119911119901minus1) isin S

lowast

119901(120574)

lArrrArr119911(119891119896lowast 119892119896)1015840

(119911)

119901isin Slowast

119901(120574)


119901(120574)

(40)


119901(120573 120574) and S(119901 120574) minusK(119901 120574)



119901120575119897and G119899119898

119901120575119897in [45] F

119901119899119897120575and

G119901119899119897120575

in [46]F119901119898119897120583

andG119901119898119897120583




Acknowledgments


References


































no 1 pp 97ndash100 2008





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












T119901((119891119896lowast 119892119896)1015840

(119911)

119901119911119901minus1) isin S

lowast

119901(120574)

lArrrArr119911(119891119896lowast 119892119896)1015840

(119911)

119901isin Slowast

119901(120574)


119901(120574)

(40)


119901(120573 120574) and S(119901 120574) minusK(119901 120574)



119901120575119897and G119899119898

119901120575119897in [45] F

119901119899119897120575and

G119901119899119897120575

in [46]F119901119898119897120583

andG119901119898119897120583




Acknowledgments


References


































no 1 pp 97ndash100 2008





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Convexity Properties for Certain Classes of … · 2019. 7. 31. · Moreover, Frasin [ ] noticed that for suitable functions % A ,theintegraloperator3 generalizes