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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 638234, 13 pages

http://dx.doi.org/10.1155/2012/638234

## Differential Subordination Results for Certain Integrodifferential Operator and Its Applications

^{1}Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura 35516, Egypt^{3}Department of Mathematics, College of Science, University of Hail, Hail, Saudi Arabia

Received 8 October 2012; Accepted 27 November 2012

Academic Editor: Josip E. Pecaric

Copyright © 2012 M. A. Kutbi and A. A. Attiya. 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.

#### Abstract

We introduce an integrodifferential operator () which plays an important role in the *Geometric Function Theory*. Some theorems in differential subordination for () are used. Applications in *Analytic Number Theory* are also obtained which give new results for Hurwitz-Lerch Zeta function and Polylogarithmic function.

#### 1. Introduction

Let denote the class of functions normalized by which are analytic in the open unit disc .

Also, let denote the class of analytic functions in the form

We begin by recalling that a general Hurwitz-Lerch Zeta function defined by (cf., e.g., [1, P. 121 et seq.]) , when , when

which contains important functions of *Analytic Number Theory*, as the Polylogarithmic function:

Several properties of can be found in the recent papers, for example Choi et al. [2], Ferreira and López [3], Gupta et al. [4], and Luo and Srivastava [5]. See, also [6–16].

Recently, Srivastava and Attiya [8] introduced the operator which makes a connection between *Geometric Function Theory* and *Analytic Number Theory*, defined by
where
and denotes the Hadamard product (or convolution).

Furthermore, Srivastava and Attiya [8] showed that As special cases of , Srivastava and Attiya [8] introduced the following identities: where, the operators and are the integral operators introduced earlier by Alexander [17] and Libera [18], respectively, is the generalized Bernardi operator, introduced by Bernardi [19], and is the Jung-Kim-Srivastava integral operator introduced by Jung et al. [20].

Moreover, in [8], Srivastava and Attiya defined the operator for , by using the following relationship:

Some applications of the operator to certain classes in *Geometric Function Theory* can be found in [21, 22].

In our investigations we need the following definitions and lemma.

*Definition 1.1. *Let and be analytic functions. The function is said to be subordinate to , written , if there exists a function analytic in , with and , and such that . If is univalent, then if and only if and .

*Definition 1.2. *Let be analytic in domain , and let be univalent in . If is analytic in with when , then we say that satisfies a first order differential subordination if
The univalent function is called dominant of the differential subordination (1.10), if for all satisfying (1.10), if for all dominant of (1.10), then we say that is the best dominant of (1.10).

Lemma 1.3 (see [8]). *If and , then
*

The purpose of the present paper is to extend the use of as integrodifferential operator, and some theorems in differential subordination for are used. Applications in *Analytic Number Theory* are also obtained which give new results for Hurwitz-Lerch Zeta function and Polylogarithmic function.

#### 2. Making Use of as a Differential Operator

From the definition of in (1.5) and using (1.7), we obtain the following identities.

For and , we have where is the Sălăgean differential operator which introduced by Sălăgean [23], is the generalized of operator, introduced by Al-Oboudi [24], was studied by Cho and Srivastava [25] and by Cho and Kim [26], and the operator was studied by Uralegaddi and Somanatha [27].

Also, we note that where is the Polylogarithmic function defined by (1.4).

Now, we prove the following lemma.

Lemma 2.1. *If and , then
**
where to -times, and denotes the composition .*

*Proof. *Putting in (1.11), we have
therefore,
Noting that the relation (2.5) is a recurrence relation, by using mathematical induction, we get (2.3), which completes the proof of the lemma.

Putting in Lemma 2.1, we obtain the following properties for both Hurwitz-Lerch Zeta function and Polylogarithmic function .

Corollary 2.2. *Let and be the Hurwitz-Lerch Zeta function and Polylogarithmic function defined by (1.3) and (1.4), respectively, then we have
**
where and .*

*Example 2.3. *Using Corollary 2.2, we have the following well known results for .(i). (ii). (iii). (iv). (v). (vi).

#### 3. Applications of Differential Subordination for

To prove our results, we need the following lemmas due to Hallenbeck and Ruscheweyh [28] and Miller and Mocanu [29], respectively, see also Miller and Mocanu [30].

Lemma 3.1. *Let be convex univalent in , with and If and
**
then
**
where
**The function is convex univalent and is the best dominant.*

Lemma 3.2. *Let , and let be the root of the equation as follows:
**In addition, let , for .**If and
**
then
*

Now, we define the function as the following:

Theorem 3.3. *Let the function defined by (3.7) and for some . If
**
then
*

The constant is the best estimate.

*Proof. *Defining the function , then we have .

If we take , and the convex univalent function defined by
then, we have
Using Lemma 1.3 and (3.7), therefore (3.11) can be written as
then,
where is defined by (3.10) satisfying .

Applying Lemma 3.1, we obtain that , where the convex univalent function defined by
Since and , we have .

This implies that
Hence, the constant cannot be replace by any larger one.

This completes the proof of Theorem 3.3.

Theorem 3.4. *Let the function with ; real, defined by (3.7), and let satisfy the following equation:
**If
**
then
*

*Proof. *Defining the function , then we have
Using Lemma 1.3 and (3.7), therefore (3.11) can be written as
This completes the proof of Theorem 3.4 after applying Lemma 3.2

#### 4. Applications in Analytic Number Theory

Putting in Theorem 3.3, then we have the following property of Hurwitz-Lerch Zeta function.

Corollary 4.1. *Let the function defined by (1.6). If
**
then
**
where and .*

The constant is the best estimate.

Putting in Theorem 3.4, then we have another property of Hurwitz-Lerch Zeta function.

Corollary 4.2. *Let the function defined by (1.6), and let satisfy the following equation:
**If
**
then
**
where and ; real.*

Putting and in Theorem 3.3, then we have the following property of Polylogarithmic function.

Corollary 4.3. *Let the function defined by
**If
**
then
**
where and .**The constant is the best estimate.*

Putting and in Theorem 3.4, then we have the following property of Polylogarithmic function.

Corollary 4.4. *Let the functions and defined by (1.6) and (4.6), respectively, and let satisfy the following:
**If
**
then
**
where and ; real.*

Setting , and in Theorem 3.3, then we have the following property of Polylogarithmic function.

Corollary 4.5. *Let the function defined by (4.6). **If
**
then
**
where and .**The constant is the best estimate.*

Taking , and in Theorem 3.4, then we have the following property of polylogarithmic function.

Corollary 4.6. *Let the function defined by (4.6). **If
**
then
**
where and .*

Corollary 4.7. *Let the function defined by (4.6) as follows: **If
**
then
**
where and .*

*Proof. *Let satisfy the condition (4.16). Also, putting , , and in Theorem 3.4.

Using (4.16), then we have
therefore
Corollary 4.5, gives
Applied (4.11) again and to -times, which gives (4.17). This completes the proof of Corollary 4.7.

Finally, we can put Corollary 4.7 in the following form.

Corollary 4.8. *Let the function defined by (4.6). **If
**
then
**
where and .*

#### Acknowledgments

This research was funded by the Deanship of Scientific Research (DSR), King Abdul-Aziz University, Jeddah, Saudi Arabia, under Grant no. 103-130-D1432. The authors, therefore, acknowledge with thanks DSR technical and financial support.

#### References

- H. M. Srivastava and J. Choi,
*Series Associated with the Zeta and Related Functions*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. - J. Choi, D. S. Jang, and H. M. Srivastava, “A generalization of the Hurwitz-Lerch zeta function,”
*Integral Transforms and Special Functions*, vol. 19, no. 1-2, pp. 65–79, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Ferreira and J. L. López, “Asymptotic expansions of the Hurwitz-Lerch zeta function,”
*Journal of Mathematical Analysis and Applications*, vol. 298, no. 1, pp. 210–224, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. L. Gupta, R. C. Gupta, S.-H. Ong, and H. M. Srivastava, “A class of Hurwitz-Lerch zeta distributions and their applications in reliability,”
*Applied Mathematics and Computation*, vol. 196, no. 2, pp. 521–531, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Q.-M. Luo and H. M. Srivastava, “Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials,”
*Journal of Mathematical Analysis and Applications*, vol. 308, no. 1, pp. 290–302, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Kutbi and A. A. Attiya, “Differential subordination result with the Srivastava-Attiya integral operator,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 618523, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Murugusundaramoorthy, “Subordination results for spiral-like functions associated with the Srivastava-Attiya operator,”
*Integral Transforms and Special Functions*, vol. 23, no. 2, pp. 97–103, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. M. Srivastava and A. A. Attiya, “An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination,”
*Integral Transforms and Special Functions*, vol. 18, no. 3-4, pp. 207–216, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Owa and A. A. Attiya, “An application of differential subordinations to the class of certain analytic functions,”
*Taiwanese Journal of Mathematics*, vol. 13, no. 2A, pp. 369–375, 2009. View at Google Scholar · View at Zentralblatt MATH - N. E. Cho, I. H. Kim, and H. M. Srivastava, “Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator,”
*Applied Mathematics and Computation*, vol. 217, no. 2, pp. 918–928, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. A. Elrifai, H. E. Darwish, and A. R. Ahmed, “Some applications of Srivastava-Attiya operator to $p$-valent starlike functions,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 790730, 11 pages, 2010. View at Publisher · View at Google Scholar - J.-L. Liu, “Sufficient conditions for strongly star-like functions involving the generalized Srivastava-Attiya operator,”
*Integral Transforms and Special Functions*, vol. 22, no. 2, pp. 79–90, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. H. Mohd and M. Darus, “Differential subordination and superordination for Srivastava-Attiya operator,”
*International Journal of Differential Equations*, vol. 2011, Article ID 902830, 19 pages, 2011. View at Publisher · View at Google Scholar - K. I. Noor and S. Z. H. Bukhari, “Some subclasses of analytic and spiral-like functions of complex order involving the Srivastava-Attiya integral operator,”
*Integral Transforms and Special Functions*, vol. 21, no. 12, pp. 907–916, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S.-M. Yuan and Z.-M. Liu, “Some properties of two subclasses of $k$-fold symmetric functions associated with Srivastava-Attiya operator,”
*Applied Mathematics and Computation*, vol. 218, no. 3, pp. 1136–1141, 2011. View at Publisher · View at Google Scholar - Z.-G. Wang, Z.-H. Liu, and Y. Sun, “Some properties of the generalized Srivastava-Attiya operator,”
*Integral Transforms and Special Functions*, vol. 23, no. 3, pp. 223–236, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,”
*Annals of Mathematics*, vol. 17, no. 1, pp. 12–22, 1915. View at Publisher · View at Google Scholar - R. J. Libera, “Some classes of regular univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 16, pp. 755–758, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. D. Bernardi, “Convex and starlike univalent functions,”
*Transactions of the American Mathematical Society*, vol. 135, pp. 429–446, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. B. Jung, Y. C. Kim, and H. M. Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,”
*Journal of Mathematical Analysis and Applications*, vol. 176, no. 1, pp. 138–147, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-L. Liu, “Subordinations for certain multivalent analytic functions associated with the generalized Srivastava-Attiya operator,”
*Integral Transforms and Special Functions*, vol. 19, no. 11-12, pp. 893–901, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. K. Prajapat and S. P. Goyal, “Applications of Srivastava-Attiya operator to the classes of strongly starlike and strongly convex functions,”
*Journal of Mathematical Inequalities*, vol. 3, no. 1, pp. 129–137, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Sălăgean, “Subclasses of univalent functions,” in
*Complex Analysis*, vol. 1013 of*Lecture Notes in Mathematics*, pp. 362–372, Springer, Berlin, Germany, 1983, Proceedings of the 5h Romanian-Finnish Seminar, Part 1, Bucharest, Romania, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. M. Al-Oboudi, “On univalent functions defined by a generalized Salagean operator,”
*International Journal of Mathematics and Mathematical Sciences*, no. 25–28, pp. 1429–1436, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. E. Cho and H. M. Srivastava, “Argument estimates of certain analytic functions defined by a class of multiplier transformations,”
*Mathematical and Computer Modelling*, vol. 37, no. 1-2, pp. 39–49, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. E. Cho and T. H. Kim, “Multiplier transformations and strongly close-to-convex functions,”
*Bulletin of the Korean Mathematical Society*, vol. 40, no. 3, pp. 399–410, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in
*Current Topics in Analytic Function Theory*, pp. 371–374, World Scientific, River Edge, NJ, USA, 1992. View at Google Scholar · View at Zentralblatt MATH - D. J. Hallenbeck and S. Ruscheweyh, “Subordination by convex functions,”
*Proceedings of the American Mathematical Society*, vol. 52, pp. 191–195, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Miller and P. T. Mocanu, “Marx-Strohhäcker differential subordination systems,”
*Proceedings of the American Mathematical Society*, vol. 99, no. 3, pp. 527–534, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Miller and P. T. Mocanu,
*Differential Subordinations: Theory and Applications*, vol. 225 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 2000.