Abstract and Applied Analysis

Volume 2013 (2013), Article ID 517296, 4 pages

http://dx.doi.org/10.1155/2013/517296

Research Article

Sufficiency Criteria for a Class of -Valent Analytic Functions of Complex Order

Muhammad Arif

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

Received 21 December 2012; Accepted 26 February 2013

Academic Editor: Fuding Xie

Copyright © 2013 Muhammad Arif. 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.

In the present paper, we consider a subclass of -valent analytic functions and obtain certain simple sufficiency criteria by using three different methods for the functions belonging to this class. Many known results appear as special consequences of our work.

1. Introduction

Let be the class of functions analytic and -valent in the open unit disk and of the form

In particular, , , and . By and , and , we mean the subclasses of which are defined, respectively, by

For , , , the previous two classes defined in (2) reduce to the well-known classes of starlike and convex, respectively.

For functions , of the form (1), we define the convolution (Hadamard product) of and by

Now we define the subclass of by

Sufficient conditions were studied by various authors for different subclasses of analytic and multivalent functions, for some of the related work see [1–8]. The object of the present paper is to obtain sufficient conditions for the subclass of . We also consider some special cases of our results which lead to various interesting corollaries and relevances of some of these with other known results also being mentioned.

We will assume throughout our discussion, unless otherwise stated, that , , .

2. Preliminary Results

To obtain our main results, we need the following Lemma's.

Lemma 1 (see [9]). If with and satisfies the condition

then

Lemma 2 (see [10]). If satisfing the condition

where is the unique root of the equation

then

Lemma 3 (see [11]). Let be a set in the complex plane , and suppose that is a mapping from to which satisfies for and for all real such that . If is analytic in and for all , then .

3. Main Results

Theorem 4. If satisfies

then .

Proof. Let us set a function by

for . Then clearly (11) shows that .

Differentiating (11) logarithmically, we have

which gives

Thus using (10), we have

Hence, using Lemma 1, we have .

From (12), we can write

Since , it implies that . Therefore, we get

and this implies that .

Setting and in Theorem 4, we get the following.

Corollary 5. If satisfies

then , the class of starlike functions of complex order .

Putting and in Theorem 4, we have the following.

Corollary 6. If satisfies

then , the class of convex functions of complex order .

Remark 7. If we put in Corollaries 5 and 6, we get the results proved by Uyanık et al. [1]. Furthermore, for , we obtain the results studied by Mocanu [2] and Nunokawa et al. [3], respectively. Also if we set with and in Theorem 4, we obtain the results due to Goyal et al. [4].

Theorem 8. If satisfies

where is the unique root of (8), then .

Proof. Let be given by (11), which clearly belongs to the class .

Now differentiating (11), we have

which gives

Thus using (19), we have

where is the root of (8). Hence, using Lemma 2, we have .

From (20), we can write

Since , it implies that . Therefore, we get (16), and hence .

Making , with and , we have the following.

Corollary 9. If satisfies

where is the unique root of (8) with , then , the class of -valent starlike functions of order .

Also if we take , with and in Theorem 8, we obtain the following result.

Corollary 10. If satisfies

where is the unique root of (8) with , then , the class of -valent convex functions of order .

Remark 11. For putting in Corollary 10 and in Corollary 9, we obtain the results proved by Mocanu [10] and Uyanık et al. [1], respectively.

Theorem 12. If satisfies

where and

then .

Proof. Let us set

Then is analytic in with .

Taking logarithmic differentiation of (28) and then by simple computation, we obtain

with

Now for all real and satisfying , we have

Reputing the values of , , , and then taking real part, we obtain

where , , are given in (27).

Let . Then and , for all real and satisfying , . Using Lemma 3, we have . This implies that

and hence .

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

Corollary 13. If satisfies

then .

Furthermore, for in Corollary 13, we have the following result proved in [13].

Corollary 14. If satisfies

then .

Acknowledgment

The author is thankful to the Prof. Dr. Ihsan Ali, Vice chancellor of Abdul Wali Khan University Mardan, for providing research facilities in AWKUM.

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