The Scientific World Journal

Volume 2014 (2014), Article ID 801751, 5 pages

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

Research Article

Coefficient Inequalities for a Subclass of p-Valent Analytic Functions

Muhammad Arif,¹ Janusz Sokół,² and Muhammad Ayaz¹

¹Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan

²Department 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

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

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