The Scientific World Journal

Volume 2014, Article ID 801751, 5 pages

http://dx.doi.org/10.1155/2014/801751

## Coefficient Inequalities for a Subclass of *p*-Valent Analytic Functions

^{1}Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan^{2}Department of Mathematics, Rzeszów University of Technology, Al. Powstancow Warszawy 12, 35-959 Rzeszów, Poland

Received 31 August 2013; Accepted 19 November 2013; Published 4 February 2014

Academic Editors: Y.-M. Chu, A. I. Delis, B. Dragovich, A. Fošner, and B. Meng

Copyright © 2014 Muhammad Arif 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.

#### Abstract

The aim of this paper is to study the problem of coefficient bounds for a newly defined subclass of *p*-valent analytic functions. Many known results appear as special consequences of our work.

#### 1. Introduction

Let denote the class of functions of the form which are analytic and multivalent in the open unit disk . Also let and denote the well-known classes of -valent starlike functions and -valent convex functions, respectively.

For given by (1) and given by the Hadamard product (or convolution) of and is given by Motivated by Ruscheweyh operator [1], Goel and Sohi [2] introduced a differential operator for -valent analytic functions given by with , and is a Pochhammer symbol given by It is obvious that when is any integer greater than , The following identity can be easily established: Using the generalized Ruscheweyh operator, we define a subclass of of -valent analytic functions as follows.

*Definition 1. *An analytic -valent function of the form (1) belongs to the class , if and only if
where , , , is real with , and .

By giving specific values to , , , , , and in , we obtain many important subclasses studied by various authors in earlier papers; see for details [3–6]; we list some of them as follows:(i) and , studied by Spacek [7] and Robertson [8], respectively; for the advancement work see [9–11];(ii) and , studied by both Owa et al. and Shams et al. [12, 13];(iii) and , introduced by Ravichandran et al. [14];(iv), considered by Latha [15];(v) and , the well-known classes of starlike and convex functions of order .From the above special cases we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.

We will assume throughout our discussion, unless otherwise stated, that , , , is real with , and .

#### 2. Main Results

Theorem 2. *Let with . Then , where .*

*Proof. *Let . Then we obtain
and this implies
Also if , then we can easily obtain
and this completes the proof.

Theorem 3. *If , then
**
and
**
where is given by (5) and
*

*Proof. *Let . Then by Theorem 2, we have
Let us define by
Then is analytic in with and , . Let
Then (17) becomes
That is,
where is given by (15). Using (8) in (20), we obtain
or, equivalently,
Comparing the coefficients of on both sides,
Taking absolute on both sides and then applying the coefficient estimates for Caratheodory functions [3], we have
We apply mathematical induction on (24). So for ,
which shows that (13) is true. For ,
and using (13), we have
Therefore, (14) holds for .

Assume that (14) is true for ; that is,
Consider
Therefore, the result is true for , and hence by using mathematical induction, (14) holds true for all .

If we put , , , and in Theorem 3, we obtain the result proved in [12].

*Corollary 4. If , then
If one takes in Corollary 4, one obtains the following inequality:
which was proved by Robertson [16].*

By setting , , , and in Theorem 3, one obtains the result proved in [12].

*Corollary 5. If , then
Letting in Corollary 5, one gets the following inequality proved by Robertson [16]:
*

*Theorem 6. If and satisfies
where is given by (5), then .*

*Proof. *Suppose (34) holds. Also let us suppose
Then
Using (8) and then simplifications gives
Now consider
The last expression is bounded by 1 if
and this completes the proof.

*For , , , and in Theorem 6, we obtain the following.*

*Corollary 7. If and satisfies
then , the class of -valent starlike functions of order .*

For , , , and in Theorem 6, one has the following.

*Corollary 8. If and satisfies
then , the class of -valent convex functions of order .*

Further for in both the last two corollaries, one obtains the results for the classes and which was proved by Merkes et al. [17] and Silverman [18], respectively.

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that they have no conflict of interests. Please consider this paper for further process.*

*References*

*References*

- S. Ruscheweyh, “A new criteria for univalent function,”
*Proceedings of the American Mathematical Society*, vol. 49, no. 1, pp. 109–115, 1975. View at Google Scholar - R. M. Goel and N. Sohi, “A new criteria for p-valent functions,”
*Proceedings of the American Mathematical Society*, vol. 78, pp. 353–357, 1980. View at Google Scholar - E. Aqlan, J. M. Jahangiri, and S. R. Kulkarni, “Classes of k-uniformly convex and starlike functions,”
*Tamkang Journal of Mathematics*, vol. 35, no. 3, pp. 1–7, 2004. View at Google Scholar - S. Kanas and A. Wisniowska, “Conic regions and k-uniform convexity,”
*Journal of Computational and Applied Mathematics*, vol. 105, no. 1-2, pp. 327–336, 1999. View at Google Scholar · View at Scopus - S. Kanas and A. Wisniowska, “Conic domains and starlike functions,”
*Revue Roumaine de Mathématique Pures et Appliquées*, vol. 45, pp. 647–657, 2000. View at Google Scholar - J. Nishiwaki and S. Owa, “Certain classes of analytic functions concerned with uniformly starlike and convex functions,”
*Applied Mathematics and Computation*, vol. 187, no. 1, pp. 350–355, 2007. View at Publisher · View at Google Scholar · View at Scopus - L. Spacek, “Prispĕvek k teorii funkei prostych,”
*Časopis pro Pěstování Matematiky a Fysik*, vol. 62, pp. 12–19, 1933. View at Google Scholar - M. S. Robertson, “Univalent functions $f(z)$ for wich $z{f}^{\text{'}}(z)$ is spiral-like,”
*Michigan Mathematical Journal*, vol. 16, pp. 97–101, 1969. View at Google Scholar - M. Arif, “On certain suffciency criteria for
*p*-valent meromorphic spiralike functions,”*Abstract and Applied Analysis*, vol. 2013, Article ID 837913, 9 pages, 2013. View at Publisher · View at Google Scholar - M. Arif, K. I. Noor, M. Raza, and W. Haq, “Some properties of a generalized class of analytic functions related with Janowski functions,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 279843, 11 pages, 2012. View at Publisher · View at Google Scholar - K. I. Noor, M. Arif, and A. Muhammad, “Mapping properties of some classes of analytic functions under an integral operator,”
*Journal of Mathematical Inequalities*, vol. 4, no. 4, pp. 593–600, 2010. View at Google Scholar - S. Owa, Y. Polatoǧlu, and E. Yavuz, “Coefficient inequalities for classes of uniformly starlike and convex functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 7, no. 5, article 160, 2006. View at Google Scholar · View at Scopus - S. Shams, S. R. Kulkarni, and J. M. Jahangiri, “Classes of uniformly starlike and convex functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2004, no. 55, pp. 2959–2961, 2004. View at Publisher · View at Google Scholar · View at Scopus - V. Ravichandran, C. Selvaraj, and R. Rajagopal, “On uniformly convex spiral functions and uniformly spirallike function,”
*Soochow Journal of Mathematics*, vol. 29, no. 4, pp. 392–405, 2003. View at Google Scholar - S. Latha, “Coefficient inequalities for certain classes of ruscheweyh type analytic functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 2, article 52, 2008. View at Google Scholar · View at Scopus - M. S. Robertson, “On the theory of univalent functions,”
*Annals of Mathematics*, vol. 37, pp. 374–408, 1936. View at Google Scholar - E. P. Merkes, M. S. Robertson, and W. T. Scott, “On products of starlike functions,”
*Proceedings of the American Mathematical Society*, vol. 13, pp. 960–964, 1962. View at Google Scholar - H. Silverman, “Univalent functions with negative coeffcients,”
*Proceedings of the American Mathematical Society*, vol. 51, pp. 109–116, 1975. View at Google Scholar

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