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We introduce and study some new subclasses of p-valent starlike, convex, close-to-convex, and quasi-convex functions defined by certain Srivastava-Attiya operator.. Goodman, “On the Schw

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Volume 2010, Article ID 790730, 11 pages

doi:10.1155/2010/790730

Research Article

Some Applications of Srivastava-Attiya Operator to

p-Valent Starlike Functions

E A Elrifai, H E Darwish, and A R Ahmed

Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Correspondence should be addressed to H E Darwish,darwish333@yahoo.com

Received 25 March 2010; Accepted 14 July 2010

Academic Editor: Ram N Mohapatra

Copyrightq 2010 E A Elrifai et al 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 and study some new subclasses of p-valent starlike, convex, close-to-convex, and

quasi-convex functions defined by certain Srivastava-Attiya operator Inclusion relations are established, and integral operator of functions in these subclasses is discussed

1 Introduction

Let Ap denote the class of functions of the form

f z z p∞

n1

a n p z n p

p ∈ N {1, 2, 3, }, 1.1

which are analytic and p-valent in the open unit disc U {z : z ∈ C and |z| < 1} Also, let the

Hadamard product orconvolution of two functions

fj z z p∞

n1

an p,j z n p

be given byf1∗ f2z z p∞

n1a n p,1 a n p,2 z n p f2∗ f1z.

A function fz ∈ Ap is said to be in the class S∗

p α of p-valent functions of order α

if it satisfies

Re

zfz

f z

> α

0≤ α < p, z ∈ U. 1.3

we write S∗p 0 S∗

p , the class of p-valent starlike in U.

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A function f ∈ Ap is said to be in the class C p α of p-valent convex functions of order α if it satisfies

Re

1zf

z

fz

> α

0≤ α < p, z ∈ U. 1.4

The class of p-valent convex functions in U is denoted by C p C p 0.

It follows from1.3 and 1.4 that

f z ∈ C p α iﬀ zfz

p ∈ S∗

p α 0≤ α < p. 1.5

The classes S∗p and C p were introduced by Goodman 1 Furthermore, a function

f z ∈ Ap is said to be p-valent close-to-convex of order β and type γ in U if there exists a function gz ∈ S∗

p γ such that

Re

zfz

g z

> β

0≤ β, γ < p, z ∈ U. 1.6

We denote this class by K p β, γ The class K p β, γ was studied by Aouf 2 We note

that K1β, γ Kβ, γ was studied by Libera 3

A function f ∈ Ap is called quasi-convex of order β type γ, if there exists a function

g z ∈ C p γ such that

Re zf

z

gz

> β, z ∈ U, 1.7

where 0 ≤ β, γ < p We denote this class by K∗

p β, γ Clearly fz ∈ K∗

p β, γ ⇔ zfz/p ∈

K p β, γ.The generalized Srivastava-Attiya operator J s,b f z : Ap → Ap in 4 is introduced by

J s,b f z G s,b z ∗ fz z ∈ U : b ∈ C \ Z0 {0, −1, −2, −3, }, s ∈ C, p ∈ N 1.8 where

Gs,b z 1 b s

φ z, s, b − b −s

,

φ z, s, b 1

b s z p

1 b s z1p

2 b s · · · 1.9

It is not diﬃcult to see from 1.8 and 1.9 that

Js,bf z z p∞

n1

1 b

n 1 b

s

an p z n p 1.10

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When p 1, the operator J s,bis well-known Srivastava-Attiya operator5.

Using the operator J s,b , we now introduce the following classes:

S∗p,s,b

γ

f z ∈ Ap

: J s,b f z ∈ S∗

p

γ

,

Cp,s,b

γ

f z ∈ Ap

: J s,bf z ∈ C p

γ

,

K p,s,b

β, γ

f z ∈ Ap

: J s,b f z ∈ K p

β, γ

,

K∗p,s,b

β, γ

f z ∈ Ap

: J s,bf z ∈ K∗

p

β, γ

.

1.11

In this paper, we will establish inclusion relation for these classes and investigate Srivastava-Attiya operator for these classes

We note that

1 for s σ, b p, we get Jung-Kim-Srivastava 6,7;

2 for s 1, 1 b c p, we get the generalized Libera integral operator 8,9;

3 for s −k being any negative integer, b 0, and p 1, the operator J −k,0 D k f z

was studied by S˘al˘agean10

2 Inclusion Relation

In order to prove our main results, we will require the following lemmas

Lemma 2.1 see 11 Let wzbe regular in U with w0 0 If |wz| attains its maximum value

on the circle |z| r at a given point z0∈ U, then z0wz0 kwz0, where k is a real number and

k ≥ 1.

Lemma 2.2 see 12 Let u u1 iu2, v v1  iv2, and let ψ u, v be a complex function,

ψ : D → C, D ⊂ C × C Suppose that ψ satisfies the following conditions:

i ψu, v is continuous in D,

ii 1, 0 ∈ D and Re{ψ1, 0} > 0,

iii Re{ψiu2, v1} ≤ 0 for aliu2, v1 ∈ D such that v1≤ −1 u2

2/2.

Let h z 1 c1z c1z2 · · · be analytic in U, such that hz, zhz ∈ D for z ∈ U If

Re{ψhz, zhz} > 0, z ∈ U then Re hz > 0 for z ∈ U.

Our first inclusion theorem is stated as follows

Theorem 2.3 S∗

p,s.b γ ⊂ S∗

p,s 1,b γ for any complex number s.

Proof Let f z ∈ S∗

p,s,b γ, and set

z

Js 1,b f z

J s 1,b f z − γ

p − γh z, 2.1

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where hz 1 c1z c2z2 · · · Using the identity

z

J s 1,b f zp − 1 b J s 1,b f z 1 bJ s,b f z, 2.2

we have

J s,b f z

J s 1,b f z

1

1 b

z

Js 1,b f z

J s 1,b f z − p b 1

,

Js,bf z

J s 1,b f z

1

b 1

γp − γh z − p b 1.

2.3

Diﬀerentiating 2.3, logarithmically with respect to z, we obtain

z

J s,b f z

J s,b f z − γ

p − γh z 

p − γzhz

p − γh z γ − p b 1 . 2.4 Now, from the function ψu, v, by taking u hz, v zhz in 2.4 as

ψ u, v p − γu

p − γv

p − γu γ − p b 1 , 2.5

it is easy to see that the function ψu, v satisfies condition i and ii ofLemma 2.2, in D

C − {γ − p b 1/γ − p} × C To verify condition iii, we calculate as follows:

Re

ψ iu2, v1 Re

p − γv1

p − γiu2 γ − p b 1

Re p − γ

v1

γ − p b 1− ip − γu2

p − γ2

u221− p b γ2

Re p − γ

γ − p b 1v1− ip − γ2

v1u2

p − γ2u2

21− p b γ2

p − γγ − p b 1v1

p − γ2

u2

21− p b γ2

≤ −

p − γγ − p b 11 u2

2

p − γ2

u221− p b γ2 < 0,

2.6

where v1 ≤ −1 u2

2/2 and iu2, v1 ∈ D Therefore, the function ψu, v satisfies the

conditions ofLemma 2.2

This shows that if Rehz, zhz > 0 z ∈ U, then

Rehz > 0, z ∈ U 2.7

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if f ∈ S∗

s γ, then

S∗p,s,b

γ

⊂ S∗

p,s 1,b

γ

This completes the proof ofTheorem 2.3

Theorem 2.4 C p,s,b γ ⊂ C p,s 1,b γ, for any complex number s.

Proof Consider the following:

f z ∈ C p,s,b

γ

⇐⇒ J s,b f z ∈ C p

γ

⇐⇒ z

p

J s,b f z∈ S∗

p

γ

⇐⇒ J s,b

zfz

p

∈ S∗

p

γ

⇐⇒zfz

p ∈ S∗

p,s,b

γ

⇒zfz

p ∈ S∗

p,s 1,b

γ

⇐⇒ J s 1,b

zfz

p

∈ S∗

p

γ

⇐⇒z

p

J s 1,b f z∈ S∗

p

γ

⇐⇒ J s 1,b f z ∈ C p

γ

⇐⇒ fz ∈ C p,s 1,b

γ

,

2.9

which evidently provesTheorem 2.4

Theorem 2.5 K p,s,b β, γ ⊂ K p,s 1,b β, γ, for any complex number s.

Proof Let f z ∈ K p,s,b β, γ Then, there exists a function kz ∈ S 

p γ such that

Re z

J s,b f z

g z

> β z ∈ U. 2.10

Taking the function kz which satisfies J s,b k z gz, we have kz ∈ S 

p γ and

Re{zJs,b f z/J s,b k z} > β z ∈ U.

Now, put zJ s 1,b f z/ J s 1,b k z − β p − βhz, where hz 1 c1z c2z2 · · ·

Using the identity2.2 we have

z

J s,b f z

Js,bk z

Js,b

zfz

Js,bk z

z

J s 1,b

zf

z−p − 1 bJ s 1,b

zf

z

z J s 1,b k z−p − 1 bJs 1,b k z

z

J s 1,b

zf z /J s 1,b k z −p − 1 bJ s 1,b

zf z/J s 1,b k z

z J s 1,b k z/Js 1,b k z −p − 1 b .

2.11

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Since kz ∈ S∗

p,s,b γ and S∗

p,s,b γ ⊂ S∗

p,s 1,b γ, we let zJ s 1,b k z/Js 1,b k z p − γHz 

γ, where Re H z > 0 z ∈ U thus 2.11 can be written as

z

J s,b f z

J s,b k z

z

J s 1,b

zf

z/J s 1,b k z −p − 1 bβp − βh z

p − γH z γ −p − 1 b . 2.12

Consider that

z

J s 1,b f z J s 1,b k zβp − βh z. 2.13 Diﬀerentiating both sides of 2.13, and multiplying by z, we have

z

J s 1,b

zf

z

J s 1,b k z

p − βzhz βp − βh z·p − γH z γ. 2.14 Using2.14 and 2.12, we get

z

Js,bf z

Js,bk z − β

p − βh z 

p − βzhz

p − γH z γ −p − 1 b . 2.15 Taking u hz, v zhz in 2.15, we form the function ψu, v as

ψ u, v p − βu

p − βv

p − γH z γ −p − 1 b . 2.16

It is not diﬃcult to see that ψu, v satisfies the conditions i and ii ofLemma 2.2in D

C × C To verify condition iii, we proceed as follows:

Re ψiu2, v1

p − βv1

p − γh1

x, y

 γ −p − 1 b

p − γh1

x, y

 γ 1 b − p2p − γh2

x, y2, 2.17

where Hz h1x, y ih2x, y, h1x, y and h2x, y being the functions of x and y and

Re Hz h1x, y > 0.

By putting v1 ≤ −1/21 u2

2, we have

Re ψiu2, v1 ≤ −

p − β1 u2

2

p − γh1

x, y

 γ −p − 1 b

2

p − γh1

x, y

 γ 1 b − p2p − γh2

x, y2 < 0. 2.18

Hence, Re hz > 0 z ∈ U and fz ∈ K p,s 1,b β, γ The proof ofTheorem 2.5is complete.

Theorem 2.6 K∗

p,s,b β, γ ⊂ K∗

p,s 1,b β, γ for any complex number s.

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f z ∈ K∗

p,s,b

β, γ

⇐⇒ J s,bf z ∈ K∗

p

β, γ

⇐⇒ z

p

J s,b f z∈ K p

β, γ

⇐⇒ J s,b

zfz

p

∈ K p

β, γ

⇒zfz

p ∈ K p,s,b

β, γ

⇒ zfz

p ∈ K p,s 1,b

β, γ

⇐⇒ J s 1,b

zfz

p

∈ K p

β, γ

⇐⇒ z

p

J s 1,b f z∈ K p

β, γ

⇐⇒ J s 1,b f z ∈ K∗

p

β, γ

⇒ fz ∈ K∗

p,s 1,b

β, γ

.

2.19

The proof ofTheorem 2.6is complete

3 Integral Operator

For c > −1 and fz ∈ Ap, we recall here the generalized Bernardi-Libera-Livingston integral operator L c f z as follows

Lcf z c p

z c

z

0t c−1f tdt

z p∞

n1

c p

c p n

a n p z n p

3.1

The operator L c fz when c ∈ N {1, 2, 3, } was studied by Bernardi 13, for

c 1, L1fz was investigated earlier by Libera 14 Now, we have

J s,b

L c f z z p∞

n1

1 b

1 b n

s c p

c p n

a n p z n p , 3.2

so we get the identity

z

Js,b

Lcf zc pJs,bf z − cLcf z. 3.3

The following theorems deal with the generalized Bernard-Libera-Livingston integral

operator L c fz defined by 3.1

Theorem 3.1 Let c > −γ, 0 ≤ γ < p If fz ∈ S∗

p,s,b γ, then L c f z ∈ S∗

p,s,b γ.

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Proof From3.3, we have

z

Js,b

Lcf z

J s,b L c f z

c pJs,bf z

Js,b

Lcf z − c

11− 2γω z

1− ωz , 3.4 where wz is analytic in U, w0 0 Using 3.3 and 3.4 we get

Js,bf z

Js,bLcf z

c p wz1− c − 2γω z

c p1 − ωz . 3.5

Diﬀerentiating 3.5, we obtain

z

Js,bf z

11− 2γw z

1− wz −

zwz

1− wz

1− c − 2γzwz

p c 1− c − 2γw z . 3.6

Now we assume that|wz| < 1 z ∈ U Otherwise, there exists a point z0 ∈ U such that

max|wz| |wz0| 1 Then byLemma 2.1, we have z0wz0 kwz0, k ≥ 1 Putting

z z0and wz0 e iθin3.6, we have

Re z0

J s,b f z0

J s,b f z0 − γ

1− γke iθ

1− e iθ

p c 1− c − 2γe iθ

1− γc γ

1 c2 21 c1− c − 2γcos θ1− c − 2γ2 ≤ 0,

3.7

which contradicts the hypothesis that fz ∈ S∗

p,s,b γ.

Hence,|wz| < 1, for z ∈ U, and it follows 3.4 that L cf ∈ S∗

p,s,b γ.

The proof ofTheorem 3.1is complete·

Theorem 3.2 Let c > −γ, 0 ≤ γ < p If f ∈ C p,s,b γ, then L c f z ∈ C p,s,b γ.

f z ∈ C p,s,b

γ

⇐⇒ zfz

p ∈ S∗

p,s,b

γ

⇒ L c

zfz

p

∈ S∗

p,s,b

γ

⇐⇒ z

p

Lcf z∈ S∗

p,s,b

γ

⇐⇒ L cf z ∈ C p,s,b

γ

.

3.8

This completes the proof ofTheorem 3.2

Theorem 3.3 Let c > −γ, 0 ≤ γ < p.If fz ∈ K p,s,b β, γ then L c fz ∈ K p,s,b β, γ.

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Proof Let f z ∈ K p,s,b β, γ Then, by definition, there exists a function gz ∈ S∗

p,s,b γ such

that

Re z

J s,b f z

Js,bg z

> β z ∈ U. 3.9 Then,

z

J s,b L c f z

J s,b L c g z − β

p − βh z 3.10

where hz c1z c2z2 · · · From 3.3 and 3.10, we have

z

J s,b f z

J s,b g z

J s,b

zfz

J s,b g z

z

J s,b L c

zfz cJ s,b L c

zf

z

J s,b L c

g z cJ s,b L c g z

z

J s,b L c zfz/J s,b L c

g z cJ s,b L c

zfz/J s,b L c

g z

z

J s,b L c g z/J s,b L c

g z c .

3.11

Since gz ∈ S∗

p,s,b γ, then fromTheorem 3.1, we have L c g ∈ S∗

p,s,b γ.

Let

z

J s,b L c

g z

J s,b L c

g z

p − γH z γ, 3.12

where Re Hz > 0z ∈ U Using 3.11, we have

z

J s,b f z

J s,b g z

z

J s,b L c

zfz/J s,b L c

g

 cp − βh z β

p − γH z γ c . 3.13

Also,3.10 can be written as

z

J s,b L c

f z J s,b L c

g zp − βh z β. 3.14 Diﬀerentiating both sides, we have

z

J s,b L c f z zJ s,b L c g zp − βh z βp − βzhzJ s,b L c g z, 3.15 or

z

J s,b L c f z

Js,bLc

g z

z

Js,bLc

zfz

Js,bLc

g z

p − βzhz p − βh z β1− γH z γ.

3.16

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Now, from3.13 we have

z

J s,b f z

Js,bg z − β

p − βh z 

p − βzhz

p − γH z γ c . 3.17

We form the function ψu, v by taking u hz, v zhz in 3.17 as follows

ψ u, v p − βu

p − βv

p − γH z γ c . 3.18

It is clear that the function ψu, v defined in D C × C by 3.18 satisfies conditions

i and ii ofLemma 2.2 To verify the conditioniii, we proceed as follows:

Re ψiu2, v1

p − βv1

p − γh1

x, y

 γ c

p − γh1

x, y

 γ c2p − γh2

x, y2, 3.19

where Hz h1x, y ih2x, y, h1x, y and h2x, y being the functions of x and y and

Re Hz h1x, y > 0.

By putting v1≤ −1/21 u2

2, we have

Re ψiu2, v1 ≤ −

p − β1 u2

2

p − γh1

x, y

 γ c

2

p − γh1

x, y

 γ c2p − γh2

x, y2 < 0. 3.20

Hence, Re hz > 0z ∈ U and L cf z ∈ K p,s,b β, γ Thus, we have L cf z ∈

Kp,s,b β, γ The proof ofTheorem 3.3is complete

Theorem 3.4 Let c > −γ, 0 ≤ γ < p If fz ∈ K∗

p,s,b β, γ, then L c f z ∈ K∗

p,s,b β, γ.

f z ∈ K∗

p,s,b

β, γ

⇐⇒ zfz ∈ K p,s,,b

β, γ

⇒ L c

zfz∈ K p,s,b

β, γ

⇐⇒ zLcf z∈ K p,s,b

β, γ

⇐⇒ L c f z ∈ K∗

p,s,b

β, γ

,

3.21

and the proof ofTheorem 3.4is complete

Acknowledgement

The authors would like to thank the referees of the paper for their helpful suggestions

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When p 1, the operator J s,bis well-known Srivastava-Attiya operator 5.

Using... families of integral operators,” Journal of Mathematical Analysis and Applications,

vol 176, no 1, pp 138–147, 1993

8 J.-L Liu, ? ?Some applications of certain integral operator, ”... class="text_page_counter">Trang 11

1 A W Goodman, “On the Schwarz-Christoﬀel transformation and p-valent functions,” Transactions of the American

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