Open Access

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

Journal of Inequalities and Applications20102010:790730

https://doi.org/10.1155/2010/790730

Received: 25 March 2010

Accepted: 14 July 2010

Published: 1 August 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
(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 S l gean [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.

Declarations

Acknowledgement

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

Authors’ Affiliations

(1)
Faculty of Science, Mansoura University, Mansoura, Egypt

References

  1. Goodman AW: On the Schwarz-Christoffel transformation and -valent functions. Transactions of the American Mathematical Society 1950, 68: 204–223.MathSciNetMATHGoogle Scholar
  2. Aouf MK: On a class of -valent close-to-convex functions of order and type . International Journal of Mathematics and Mathematical Sciences 1988, 11(2):259–266. 10.1155/S0161171288000316MathSciNetView ArticleMATHGoogle Scholar
  3. Libera RJ: Some radius of convexity problems. Duke Mathematical Journal 1964, 31(1):143–158. 10.1215/S0012-7094-64-03114-XMathSciNetView ArticleMATHGoogle Scholar
  4. Liu J-L: Subordinations for certain multivalent analytic functions associated with the generalized Srivastava-Attiya operator. Integral Transforms and Special Functions 2008, 19(11–12):893–901.MathSciNetView ArticleMATHGoogle Scholar
  5. Srivastava HM, Attiya AA: An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination. Integral Transforms and Special Functions 2007, 18(3–4):207–216.MathSciNetView ArticleMATHGoogle Scholar
  6. Liu J-L: Notes on Jung-Kim-Srivastava integral operator. Journal of Mathematical Analysis and Applications 2004, 294(1):96–103. 10.1016/j.jmaa.2004.01.040MathSciNetView ArticleMATHGoogle Scholar
  7. Jung IB, Kim YC, Srivastava HM: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. Journal of Mathematical Analysis and Applications 1993, 176(1):138–147. 10.1006/jmaa.1993.1204MathSciNetView ArticleMATHGoogle Scholar
  8. Liu J-L: Some applications of certain integral operator. Kyungpook Mathematical Journal 2003, 43(2):211–219.MathSciNetMATHGoogle Scholar
  9. Saitoh H: A linear operator and its applications to certain subclasses of multivalent functions. Sūrikaisekikenkyūsho Kōkyūroku 1993, (821):128–137.MathSciNetGoogle Scholar
  10. Sălăgean GŞ: Subclasses of univalent functions. In Complex Analysis, Lecture Notes in Mathematics. Volume 1013. Springer, Berlin, Germany; 1983:362–372. 10.1007/BFb0066543Google Scholar
  11. Jack IS: Functions starlike and convex of order . Journal of the London Mathematical Society 1971, 3: 469–474. 10.1112/jlms/s2-3.3.469MathSciNetView ArticleMATHGoogle Scholar
  12. Miller SS, Mocanu PT: Second-order differential inequalities in the complex plane. Journal of Mathematical Analysis and Applications 1978, 65(2):289–305. 10.1016/0022-247X(78)90181-6MathSciNetView ArticleMATHGoogle Scholar
  13. Bernardi SD: Convex and starlike univalent functions. Transactions of the American Mathematical Society 1969, 135: 429–446.MathSciNetView ArticleMATHGoogle Scholar
  14. Libera RJ: Some classes of regular univalent functions. Proceedings of the American Mathematical Society 1965, 16: 755–758. 10.1090/S0002-9939-1965-0178131-2MathSciNetView ArticleMATHGoogle Scholar

Copyright

© E. A. Elrifai et al. 2010

This article is published under license to BioMed Central Ltd. 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.