On Starlike and Convex Functions with Respect to

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International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 834064,9pages

doi:10.1155/2011/834064

Research Article

On Starlike and Convex Functions with Respect to

k

-Symmetric Points

Afaf A. Ali Abubaker and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600 Selangor, Malaysia

Correspondence should be addressed to Maslina Darus,maslina@ukm.my

Received 29 January 2011; Revised 13 March 2011; Accepted 19 March 2011

Academic Editor: Stanisława R. Kanas

Copyrightq2011 A. A. A. Abubaker and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce new subclassesSσ,s_k λ, δ, φandK_kσ,sλ, δ, φof analytic functions with respect tok -symmetric points defined by diﬀerential operator. Some interesting properties for these classes are obtained.

1. Introduction

LetAdenote the class of functions of the form

fz z

∞

n2

anzn, 1.1

which are analytic in the unit diskU{z∈ :|z|<1}.

Also let℘be the class of analytic functions pwith p0 1, which are convex and univalent inUand satisfy the following inequality:

Ê

pz>0, z∈U. 1.2

A functionf ∈Ais said to be starlike with respect to symmetrical points inUif it satisfies

Ê

_zf_z

fz−f−z

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This class was introduced and studied by Sakaguchi in 19591. Some related classes are studied by Shanmugam et al.2.

In 1979, Chand and Singh 3defined the class of starlike functions with respect to

k-symmetric points of orderα0≤α <1. Related classes are also studied by Das and Singh

4.

Recall that the functionFis subordinate toGif there exists a functionω, analytic inU, withω0 0 and|ωz|<1, such thatFz Gwz,z∈U. We denote this subordination byFz≺Gz. IfGzis univalent inU, then the subordination is equivalent toF0 G0 andFU⊂GU.

A functionf ∈Ais in the classSkφsatisfying

zfz

fkz ≺φz,

z∈U, 1.4

whereφ∈℘,kis a fixed positive integer, andfkzis given by the following:

fkz 1

k

k−1

ν0

ε−νfενz

z

∞

ι2

akι−11zkι−11,

εexp

2πi k

, z∈U

.

1.5

The classes Skφ of starlike functions with respect to k-symmetric points and Kkφ of

convex functions with respect tok-symmetric points were considered recently by Wang et al.

5. Moreover, the special case

φz 1βz

1−αβz, 0≤α≤1, 0< β≤1 1.6

imposes the classSkα, β, which was studied by Gao and Zhou6, and the classS1φ

S∗φwas studied by Ma and Minda7.

Let two functions given byfz z ∞_n2anznandgz z ∞_n2bnznbe analytic

inU. Then the Hadamard productor convolutionf∗gof the two functionsf,gis defined by

fz∗gz z

∞

n2

anbnzn, 1.7

and for several functionf1z, . . . , fmz∈A,

f1z∗ · · · ∗fmz z

∞

n2

a1n· · ·amnzn, z∈U. 1.8

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We now define diﬀerential operator as follows:

D_λ,δσ,sfz z

∞

n2

nsCδ, n1λn−1σanzn, 1.9

where λ ≥ 0, Cδ, n δ 1_n₋₁/n−1!, for δ, σ, s ∈ N0 {0,1,2, . . .}, and xn is the

Pochhammer symbol defined by

x_n Γxn Γx

⎧ ⎨ ⎩

1, n0,

xx1· · ·xn−1, n{1,2,3, . . .}.

1.10

HereD_λ,δσ,sfzcan also be written in terms of convolution as

ψz

λz 1−z2 −

λz

1−z z

1−z

∗ z

1−zδ1, z∈U,

Dσ,s_λ,δfz ψz∗ · · · ∗ψz

σ-times

∗∞

n1

nszn∗fz Dδ∗ · · · ∗ Dδ

σ-times

∗Dσ,s_λ fz,

1.11

whereDδz ∞n2Cδ, nznandDλσ,sz

∞

n2ns1λn−1σzn.

Note thatD0,1_λ,δfz D_1,01,0fz z fzandD_λ,δ0,0fz fz. Whenσ 0, we get the S ˇul ˇugean diﬀerential operator9, whenλs0,σ1 we obtain the Ruscheweyh operator

8, whens 0,σ 1, we obtain the Al-Shaqsi and Darus 11, and when δ s 0, we obtain the Al-Oboudi diﬀerential operator10.

In this paper, we introduce new subclasses of analytic functions with respect to k -symmetric points defined by diﬀerential operator. Some interesting properties ofSσ,s_k λ, δ, φ

andK_kσ,sλ, δ, φare obtained.

Applying the operatorD_λ,δσ,sfz

Dσ,s_λ,δfkz

1

k

k−1

ν0

ε−νDσ,s_λ,δfενz, εk 1, 1.12

wherekis a fixed positive integer, we now define classes of analytic functions containing the diﬀerential operator.

Definition 1.1. LetSσ,s_k λ, δ, φdenote the class of functions inAsatisfying the condition

zDσ,s_λ,δfz Dσ,s_λ,δfkz

≺φz, 1.13

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Definition 1.2. Let K_kσ,sλ, δ, φdenote the class of functions inAsatisfying the condition

zDσ,s_λ,δfz

Dσ,s_λ,δfkz

≺φz, 1.14

whereφ∈℘.

In order to prove our results, we need the following lemmas.

Lemma 1.3see13. Letc >−1, and letIc:A → Abe the integral operator defined byF Icf,

where

Fz c1

zc

_z

0

tc−1ftdt. 1.15

Letφbe a convex function, withφ0 1 andÊ{φz c} >0 inU. Iff ∈Aandzf

_z/f_z_≺

φz, thenzFz/Fz≺qz≺φz, whereqis univalent and satisfies the diﬀerential equation

qz _qzzqz

c φz. 1.16

Lemma 1.4see14. Letκ,υbe complex numbers. Letφbe convex univalent inUwithφ0 1

andÊκφυ>0,z∈U, and letqz∈Awithq0 1 andqz≺ φz. Ifpz 1p1z

p2z2· · · ∈℘withp0 1, then

pz _κqzzpz

υ ≺φz ⇒pz≺φz. 1.17

Lemma 1.5 see 15. Letf and g, respectively, be in the classes convex function and starlike

function. Then, for every functionH∈A, one has

fz∗gzHz

fz∗gz ∈coHU, z∈U, 1.18

where coHUdenotes the closed convex hull ofHU.

2. Main Results

Theorem 2.1. Letf ∈Sσ,s_k λ, δ, φ. Thenfkdefined by1.5is inSσ,s₁ λ, δ, φ Sσ,sλ, δ, φ.

Proof. Letf∈Sσ,s_k λ, δ, φ, then byDefinition 1.1we have

zDσ,s_λ,δfz Dσ,s_λ,δfkz

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Substitutingzbyεν_z_{, where}_εk₁_ν₀_,₁_{, . . . , k}₋₁_in_2.1_{, respectively, we have}

εν_z_Dσ,s

λ,δfενz

Dσ,s_λ,δfkενz

≺φz. 2.2

According to the definition offkandεk 1, we knowfkενz ενfkzfor anyν0,1, . . . ,

k−1, and summing up, we can get

1

k

k−1

ν0

ε−ν

⎡ ⎢

⎣z

Dσ,s_λ,δfεν_z

Dσ,s_λ,δfkz ⎤ ⎥

⎦ z

1/k k_ν0−1ε−ν_Dσ,s

λ,δfενz

Dσ,s_λ,δfkz

z

Dσ,s_λ,δfkz

D_λ,δσ,sfkz

. 2.3

Hence there existζνinUsuch that

zDσ,s_λ,δfkz

Dσ,s_λ,δfkz

1 k

k−1

ν0

φζν φζ0, 2.4

forζ0 ∈UsinceφUis convex. Thusfk∈Sσ,sλ, δ, φ.

Theorem 2.2. Letf ∈Aandφ∈℘. Then

f∈K_kσ,sλ, δ, φ⇐⇒zf∈Sσ,s_k λ, δ, φ. 2.5

Proof. Let

gz z

∞

n2

ns_{Cδ, n}₁_λn₋₁σ_zn_, _2.6

and the operatorDσ,s_λ,δfcan be written asDσ,s_λ,δf g∗f.

Then from the definition of the diﬀerential operatorDσ,s_λ,δ, we can verify

zDσ,s_λ,δfz

Dσ,s_λ,δfkz

zg∗fz

g∗f_kz

zg∗zfz

g∗zf_kz

zD_λ,δσ,szfz

Dσ,s_λ,δzf_kz . 2.7

Thusf∈Kσ,s_k λ, δ, φif and only ifzf∈Sσ,s_k λ, δ, φ.

By using Theorems2.2and2.1, we get the following.

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Proof. Letf ∈K_kσ,sλ, δ, φ. ThenTheorem 2.2shows thatzf∈Sσ,s_k λ, δ, φ. We deduce from

Theorem 2.1thatzf_k ∈Sσ,s_{λ, δ, φ}_{. From}_zf

k zfk,Theorem 2.2now shows thatfk ∈

K₁σ,sλ, δ, φ Kσ,s_{λ, δ, φ}_.

Theorem 2.4. Letφ∈℘,λ >0 withÊφz 1/λ−1>0. Iff∈S

σ,s

k λ, δ, φ, then

zDσ_λ,δ−1,sDδfkz

D_λ,δσ−1,sDδfkz

≺qz≺φz, 2.8

whereDσ_λ,δ−1,sDδfkz D_λ,δσ−1,s∗Dδfkzandqis the univalent solution of the diﬀerential equation

qz _qzzqz

1/λ−1 φz. 2.9

Proof. Letf∈Sσ,s_k λ, δ, φ. Then in view ofTheorem 2.1,fk∈Sσ,sλ, δ, φ, that is,

zDσ,s_λ,δfkz

Dσ,s_λ,δfkz

≺φz. 2.10

From the definition ofDσ,s_λ,δ, we see that

Ds,σ_λ,δfkz 1−λ

D_λ,δs,σ−1∗Dδfkz

λzDs,σ_λ,δ−1∗Dδfkz

2.11

which implies that

Ds,σ_λ,δ−1∗Dδfkz 1

λz1/λ−1

_z

0

t1/λ−2D_λ,δs,σfktdt. 2.12

Using2.10and2.12, we see thatLemma 1.3can be applied to get2.8, wherec 1/λ− 1>−1 andÊ{φ}>0 withÊφz 1/λ−1>0 andqsatisfies2.9. We thus complete the

proof ofTheorem 2.4.

Theorem 2.5. Letφ∈℘ands∈N0. Then

Sσ,s1_k λ, δ, φ⊂Sσ,s_k λ, δ, φ. 2.13

Proof. Letf∈Sσ,s1_k λ, δ, φ. Then

zD_λ,δσ,s1fz

Dσ,s1_λ,δ fkz

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Set

pz z

Dσ,s_λ,δfz Dσ,s_λ,δfkz

, 2.15

wherepis analytic function withp0 1. By using the equation

zDσ,s_λ,δfzDσ,s1_λ,δ fz, 2.16

we get

pz D

σ,s1

λ,δ fz

Dσ,s_λ,δfkz

2.17

and then diﬀerentiating, we get

zD_λ,δσ,s1fzzDσ,s_λ,δfkzpz z

Dσ,s_λ,δfkz

pz, 2.18

Hence

zDσ,s1_λ,δ fz Dσ,s1_λ,δ fkz

D

σ,s

λ,δfkz

Dσ,s1_λ,δ fkz

zpz z

Dσ,s_λ,δfkz

Dσ,s1_λ,δ fkz

pz. 2.19

Applying2.16for the functionfkwe obtain

zD_λ,δσ,s1fz

D_λ,δσ,s1fkz

D

σ,s

λ,δfkz

Dσ,s1_λ,δ fkz

zpz pz. 2.20

Using2.20withqz D_λ,δσ,s1fkz/Dσ,s_λ,δfkz, we obtain

zDσ,s1_λ,δ fz Dσ,s1_λ,δ fkz

zpz

qz pz. 2.21

Sincef ∈Sσ,s1_k λ, δ, φ, then by using2.14in2.21we get the following.

zpz

qz pz≺φz. 2.22

We can see thatqz≺φz, hence applyingLemma 1.4we obtain the required result.

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Corollary 2.6. Letφ∈℘ands∈N0. Then

Kσ,s1_k λ, δ, φ⊂Kσ,s_k λ, δ, φ. 2.23

Now we prove that the classSσ,s_k λ, δ, φ,φ ∈ ℘, is closed under convolution with convex functions.

Theorem 2.7. Letf ∈Sσ,s_k λ, δ, φ,φ ∈ ℘, andϕis a convex function with real coeﬃcients inU.

Thenf∗ϕ∈Sσ,s_k λ, δ, φ.

Proof. Letf ∈Sσ,s_k λ, δ, φ, thenTheorem 2.1asserts thatDσ,s_λ,δfkz∈S∗φ, whereÊ{φ}>0.

ApplyingLemma 1.5and the convolution properties we get

zD_λ,δσ,sf∗ϕz D_λ,δσ,sfk∗ϕ

z

zDσ,s_λ,δfz∗ϕz ϕz∗Dσ,s_λ,δfkz

ϕz∗

zD_λ,δσ,sfz/fDσ,s_λ,δfkz

D_λ,δσ,sfkz

ϕz∗Dσ,s_λ,δfkz

∈co

⎛ ⎜

⎝z

Dσ,s_λ,δf D_λ,δσ,sfk

U

⎞ ⎟

⎠⊆φU.

2.24

Corollary 2.8. Letf ∈K_kσ,sλ, δ, φ,φ∈℘, andϕis a convex function with real coeﬃcients inU.

Thenf∗ϕ∈K_kσ,sλ, δ, φ.

Proof. Letf ∈K_kσ,sλ, δ, φ,φ∈℘. ThenTheorem 2.2shows thatzf∈Sσ,s_k λ, δ, φ. The result

ofTheorem 2.7yieldszf∗ϕzf∗ϕ∈Sσ,s_k λ, δ, φ, and thusf∗ϕ∈Kσ,s_k λ, δ, φ.

Some other works related to other diﬀerential operators with respect to symmetric points for diﬀerent types of problems can be seen in16–21.

Acknowledgments

The work presented here was partially supported by UKM-ST-06-FRGS0244-2010, and the authors would like to thank the anonymous referees for their informative and critical comments on the paper.

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On Starlike and Convex Functions with Respect to

Research Article

On Starlike and Convex Functions with Respect to

k

-Symmetric Points

Afaf A. Ali Abubaker and Maslina Darus

1. Introduction

2. Main Results

Acknowledgments

References

Submit your manuscripts at

http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific

World Journal

Discrete Mathematics

Stochastic Analysis