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Some Applications of Srivastava-Attiya Operator to p-Valent Starlike Functions
Journal of Inequalities and Applications volume 2010, Article number: 790730 (2010)
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
which are analytic and p-valent in the open unit disc Also, let the Hadamard product or (convolution) of two functions
be given by
A function is said to be in the class of p-valent functions of order if it satisfies
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
The class of p-valent convex functions in is denoted by
It follows from (1.3) and (1.4) that
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
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
where We denote this class by . Clearly The generalized Srivastava-Attiya operator in [4] is introduced by
where
It is not difficult to see from (1.8) and (1.9) that
When the operator is well-known Srivastava-Attiya operator [5].
Using the operator we now introduce the following classes:
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
where Using the identity
we have
Differentiating (2.3), logarithmically with respect to , we obtain
Now, from the function by taking in (2.4) as
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:
where and Therefore, the function satisfies the conditions of Lemma 2.2.
This shows that if then
if then
This completes the proof of Theorem 2.3.
Theorem 2.4.
, for any complex number s.
Proof.
Consider the following:
which evidently proves Theorem 2.4.
Theorem 2.5.
, for any complex number s.
Proof.
Let Then, there exists a function such that
Taking the function which satisfies we have and
Now, put where Using the identity (2.2) we have
Since and , we let , where thus (2.11) can be written as
Consider that
Differentiating both sides of (2.13), and multiplying by , we have
Using (2.14) and (2.12), we get
Taking in (2.15), we form the function as
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:
where and being the functions of and and
By putting , we have
Hence, and The proof of Theorem 2.5 is complete
Theorem 2.6.
for any complex number s.
Proof.
Consider the following:
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
The operator when was studied by Bernardi [13], for was investigated earlier by Libera [14]. Now, we have
so we get the identity
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
where is analytic in , Using (3.3) and (3.4) we get
Differentiating (3.5), we obtain
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
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:
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
Then,
where From (3.3) and (3.10), we have
Since then from Theorem 3.1, we have
Let
where Using (3.11), we have
Also, (3.10) can be written as
Differentiating both sides, we have
or
Now, from (3.13) we have
We form the function by taking in (3.17) as follows
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:
where and being the functions of and and
By putting , we have
Hence, and . Thus, we have The proof of Theorem 3.3 is complete.
Theorem 3.4.
Let If , then
Proof.
Consider the following:
and the proof of Theorem 3.4 is complete.
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The authors would like to thank the referees of the paper for their helpful suggestions.
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Elrifai, E., Darwish, H. & Ahmed, A. Some Applications of Srivastava-Attiya Operator to p-Valent Starlike Functions. J Inequal Appl 2010, 790730 (2010). https://doi.org/10.1155/2010/790730
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DOI: https://doi.org/10.1155/2010/790730