Chinese Journal of Mathematics

Chinese Journal of Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 974175 | 7 pages | https://doi.org/10.1155/2014/974175

Inclusion Properties of New Classes of Analytic Functions

Academic Editor: Yuqiang Feng
Received17 Oct 2013
Accepted24 Nov 2013
Published31 Mar 2014

Abstract

The purpose of the present paper is to introduce certain new subclasses of analytic functions defined by Srivastava-Attiya operator and study their inclusion relationships and to obtain some interesting consequences of the inclusion relations.

1. Introduction, Definitions, and Preliminaries

Let be the class of analytic functions defined on the unit disk , normalized by the conditions . Let be the subclass of consisting of univalent functions. Let denote the class of analytic convex univalent functions defined in satisfying the conditions and for all .

For any two analytic functions and defined in the unit disc , we say that is subordinate to in , denoted by , if is univalent in , and .

The convolution or the Hadamard product of two analytic functions with power series representations convergent in is the function , with power series representation The following classes of functions were defined by Shanmugam [1].

For a fixed analytic function and , We note that .

When and , , the class of starlike functions; , the class of convex functions; and , the class of close to convex functions.

The generalized Hurwitz-Lerch Zeta function is defined as where   .

This function contains, as its special cases, functions such as the Riemann and Hurwitz zeta function, Lerch zeta function, the polylogarithmic function, and the Lipschitz Lerch zeta function.

Several interesting properties and characteristics of the Hurwitz-Lerch zeta function can be found in the recent investigations by Choi and Srivastava [2], Ferreira and López [3], Garg et al. [4], Lin and Srivastava [5], and Lin et al. [6].

Using this function, Srivastava and Attiya [7] introduced the following family of linear operator , defined by where the function is given by Using (6) in (5), we get For and , where denotes Alexander transform [8] and denotes Bernardi integral operator [9].

By (7), Srivastava and Attiya obtained the following relation: In order to obtain our main results, we need the following lemmas.

Lemma 1 (see [10]). Let and be constants. Let be convex univalent in with and for in . Let be analytic in with . Then

Lemma 2 (see [11]). Let and be constants. Let be convex univalent in with and for in . Let be analytic in with . Let be analytic in with . Then

Lemma 3 (see [12]). Let a function be analytic in and . If there exists a point such that and , then where and .

2. Main Results

First we will define a subclass as follows.

Definition 4. For a fixed analytic function and a real ,

Remark 5. When , , and , reduces to the class .

Theorem 6. For ,   .

Proof. Let and .
Note that is analytic and .
Then, Hence where Since .
Thus, by using Lemma 2, That is, .

In the following, we obtain an inclusion relation of the class .

Theorem 7. For .

Proof. Let .
Then and by Theorem 6, Thus, Since is a convex domain, where .
That is, Let . Then and (23) implies that .

Theorem 8. Let . If then , for .

Proof. Let .
We prove the result by mathematical induction on .
When , .
When , set Clearly, is analytic and .
From (9), we get By logarithmic differentiation and multiplication by , we get As , by Lemma 1, we get .
That is, .
Suppose that .
Set Clearly, is analytic and .
From (9), we get By logarithmic differentiation and multiplication by , we get Hence, by Lemma 1, , for .
That is, , which proves the result.

Theorem 9. Let . If then , for .

Proof. Let .
Then

Remark 10. The Bernardi integral operator preserves both the classes and whenever .

Corollary 11. Let , , and .(i)If then .(ii)If then .

Theorem 12. Let and Then

Proof. Let .
From (9), we get From Theorem 8, for all.
Hence,
Let Then for , This completes the proof.

In the following, we introduce certain classes of analytic functions defined by Srivastava-Attiya operator and investigate their inclusion relationships.

Let and .

Definition 13. For and , let

Definition 14. For and , let

Remark 15. Clearly, if and only if .

Definition 16. For and , let

Remark 17. We note that

Theorem 18. If , then .

Proof. Let . Then, Let where is analytic and .
Using (42) in (9), we get On logarithmic differentiation and multiplication by , we arrive at Since is convex univalent in with and , Lemma 1 and the subordination relation (44) gives proving the theorem.

Theorem 19. If , then .

Proof. Let .
Then

Theorem 20. If , then .

Proof. Let . Then, by Definition 16, there exists a function such that Using Theorem 18, we get .
Thus, where is analytic.
Let where is analytic and .
Now, making use of (9), Simplifying the above, using (48) and (49), and then using (47) we arrive at Since is convex univalent in with and , Lemma 2 and the subordination relation (51) gives proving the assertion.

Now, we consider certain interesting consequences of the above inclusion relationships.

Corollary 21. Suppose that .(i)If then .(ii)If then .(iii)If then .

Corollary 22. Suppose that and . (i)If then .(ii)If then .(iii)If then .

In a similar manner one can obtain many applications of Theorems 18, 19, and 20.

Using Srivastava-Attiya integral operator, we introduce some classes of analytic functions, analogous to the class of strongly starlike functions of order and the class of strongly convex functions of order , and study the properties of the newly defined classes.

Let , , , , and .

Definition 23. Let

Definition 24. Let

Remark 25. Clearly, if and only if .

Remark 26. For , , is the class of strongly starlike functions of order and is the class of strongly convex functions of order .

Now we prove inclusion relationships for the above defined classes.

Theorem 27. Suppose that . Then for each .

Proof. Let .
We set where is analytic in , and for all .
Using (9) and (55), we have On logarithmic differentiation and multiplication by , we get Suppose that there exists a point such that Then, applying Lemma 3, we can write Thus, if , then from (57) This implies that which contradicts the condition that .
Similarly, if arg , then we obtain that which also contradicts the condition that .
Thus we have That is,

Theorem 28. Suppose that . Then , for .

Proof. Let .
Then

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. T. N. Shanmugam, “Convolution and differential subordination,” International Journal of Mathematics and Mathematical Sciences, vol. 12, pp. 333–340, 1989. View at: Google Scholar
  2. J. Choi and H. M. Srivastava, “Certain families of series associated with the Hurwitz-Lerch Zeta function,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 399–409, 2005. View at: Publisher Site | Google Scholar
  3. 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 Site | Google Scholar
  4. M. Garg, K. Jain, and H. M. Srivastava, “Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions,” Integral Transforms and Special Functions, vol. 17, no. 11, pp. 803–815, 2006. View at: Publisher Site | Google Scholar
  5. S.-D. Lin and H. M. Srivastava, “Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 725–733, 2004. View at: Publisher Site | Google Scholar
  6. S.-D. Lin, H. M. Srivastava, and P.-Y. Wang, “Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions,” Integral Transforms and Special Functions, vol. 17, no. 11, pp. 817–827, 2006. View at: Publisher Site | Google Scholar
  7. 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, pp. 207–216, 2007. View at: Publisher Site | Google Scholar
  8. J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,” Annals of Mathematics, vol. 17, pp. 12–22, 1915-1916. View at: Google Scholar
  9. S. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol. 135, pp. 429–446, 1969. View at: Google Scholar
  10. P. Eeningenburg, S. S. Miller, P. T. Mocanu, and M. O. Reade, “On a briot boquet differential subordination,” in General Inequalities 3: 3rd International Conference on General Inequalities, vol. 64, pp. 339–348, 1983. View at: Google Scholar
  11. K. S. Padmanabhan and R. Parvatam, “Some applications of differential subordination,” Bulletin of the Australian Mathematical Society, vol. 37, pp. 321–330, 1985. View at: Google Scholar
  12. M. Nunokawa, “On the order of strongly starlikeness of strongly convex functions,” Proceedings of the Japan Academy A, vol. 69, pp. 234–237, 1993. View at: Google Scholar

Copyright © 2014 Mohan Das 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.


More related articles

610 Views | 428 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.