Some Applications of Srivastava-Attiya Operator to p-Valent Starlike Functions

Abstract

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 denote the class of functions of the form

(1.1)

which are analytic and p-valent in the open unit disc Also, let the Hadamard product or (convolution) of two functions

(1.2)

be given by

A function is said to be in the class of p-valent functions of order if it satisfies

(1.3)

we write the class of p-valent starlike in .

A function is said to be in the class of p-valent convex functions of order if it satisfies

(1.4)

The class of p-valent convex functions in is denoted by

It follows from (1.3) and (1.4) that

(1.5)

The classes and were introduced by Goodman [1]. Furthermore, a function is said to be p-valent close-to-convex of order and type in if there exists a function such that

(1.6)

We denote this class by The class was studied by Aouf [2]. We note that was studied by Libera [3].

A function is called quasi-convex of order type if there exists a function such that

(1.7)

where We denote this class by . Clearly The generalized Srivastava-Attiya operator in [4] is introduced by

(1.8)

where

(1.9)

It is not difficult to see from (1.8) and (1.9) that

(1.10)

When the operator is well-known Srivastava-Attiya operator [5].

Using the operator we now introduce the following classes:

(1.11)

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

We note that

for we get Jung-Kim-Srivastava ([6, 7]);

for we get the generalized Libera integral operator. [8, 9];

for being any negative integer, and , the operator was studied by Slgean [10].

2. Inclusion Relation

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

Lemma 2.1 (see [11]).

Let be regular in U with . If attains its maximum value on the circle at a given point then where is a real number and

Lemma 2.2 (see [12]).

Let    and let be a complex function, Suppose that satisfies the following conditions:

is continuous in D,

and

for al such that

Let be analytic in , such that for If then for

Our first inclusion theorem is stated as follows.

Theorem 2.3.

for any complex number .

Proof.

Let and set

(2.1)

where Using the identity

(2.2)

we have

(2.3)

Differentiating (2.3), logarithmically with respect to , we obtain

(2.4)

Now, from the function by taking in (2.4) as

(2.5)

it is easy to see that the function satisfies condition (i) and (ii) of Lemma 2.2, in . To verify condition (iii), we calculate as follows:

(2.6)

where and Therefore, the function satisfies the conditions of Lemma 2.2.

This shows that if then

(2.7)

if then

(2.8)

This completes the proof of Theorem 2.3.

Theorem 2.4.

, for any complex number s.

Proof.

Consider the following:

(2.9)

which evidently proves Theorem 2.4.

Theorem 2.5.

, for any complex number s.

Proof.

Let Then, there exists a function such that

(2.10)

Taking the function which satisfies we have and

Now, put where Using the identity (2.2) we have

(2.11)

Since and , we let , where thus (2.11) can be written as

(2.12)

Consider that

(2.13)

Differentiating both sides of (2.13), and multiplying by , we have

(2.14)

Using (2.14) and (2.12), we get

(2.15)

Taking in (2.15), we form the function as

(2.16)

It is not difficult to see that satisfies the conditions (i) and (ii) of Lemma 2.2 in To verify condition (iii), we proceed as follows:

(2.17)

where and being the functions of and and

By putting , we have

(2.18)

Hence, and The proof of Theorem 2.5 is complete

Theorem 2.6.

  for any complex number s.

Proof.

Consider the following:

(2.19)

The proof of Theorem 2.6 is complete.

3. Integral Operator

For and , we recall here the generalized Bernardi-Libera-Livingston integral operator as follows

(3.1)

The operator when was studied by Bernardi [13], for was investigated earlier by Libera [14]. Now, we have

(3.2)

so we get the identity

(3.3)

The following theorems deal with the generalized Bernard-Libera-Livingston integral operator defined by (3.1).

Theorem 3.1.

Let If , then

Proof.

From (3.3), we have

(3.4)

where is analytic in , Using (3.3) and (3.4) we get

(3.5)

Differentiating (3.5), we obtain

(3.6)

Now we assume that Otherwise, there exists a point such that Then by Lemma 2.1, we have Putting and in (3.6), we have

(3.7)

which contradicts the hypothesis that

Hence, , for and it follows (3.4) that

The proof of Theorem 3.1 is complete·

Theorem 3.2.

Let If then

Proof.

Consider the following:

(3.8)

This completes the proof of Theorem 3.2.

Theorem 3.3.

Let If then

Proof.

Let Then, by definition, there exists a function such that

(3.9)

Then,

(3.10)

where From (3.3) and (3.10), we have

(3.11)

Since then from Theorem 3.1, we have

Let

(3.12)

where Using (3.11), we have

(3.13)

Also, (3.10) can be written as

(3.14)

Differentiating both sides, we have

(3.15)

or

(3.16)

Now, from (3.13) we have

(3.17)

We form the function by taking in (3.17) as follows

(3.18)

It is clear that the function defined in by (3.18) satisfies conditions (i) and (ii) of Lemma 2.2. To verify the condition(iii), we proceed as follows:

(3.19)

where and being the functions of and and

By putting , we have

(3.20)

Hence, and . Thus, we have The proof of Theorem 3.3 is complete.

Theorem 3.4.

Let    If , then

Proof.

Consider the following:

(3.21)

and the proof of Theorem 3.4 is complete.