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Journal of Complex Analysis
Volume 2013 (2013), Article ID 958210, 5 pages
http://dx.doi.org/10.1155/2013/958210
Research Article

Certain New Classes of Analytic Functions with Varying Arguments

1Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 33516, Egypt
3Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Received 3 November 2012; Accepted 3 January 2013

Academic Editor: Jacek Dziok

Copyright © 2013 R. M. El-Ashwah 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

We introduce certain new classes and , which represent the κ uniformly starlike functions of order α and type β with varying arguments and the κ uniformly convex functions of order α and type β with varying arguments, respectively. Moreover, we give coefficients estimates, distortion theorems, and extreme points of these classes.

1. Introduction

Let denote the class of functions of the following form: that are analytic and univalent in the open unit disc .

Definition 1 (see [1]). Let denote the subclass of consisting of functions of the form (1) and satisfy the following inequality: Also let denote the subclass of consisting of functions of the form (1) and satisfy the following inequality: It follows from (2) and (3) that The class denote the class of uniformly starlike functions of order and type and the class denotes the class of uniformly convex functions of order and type .

Specializing parameters ,  , and , we obtain the following subclasses studied by various authors:(i) and (see [2, 3]);(ii) and (see [4]);(iii) and (see [5, 6]);(iv) and (see [4, 710]).

Also we note that which are the uniformly starlike functions of order and type and uniformly convex functions of order and type , respectively.

Definition 2 (see [11]). A function of the form (1) is said to be in the class if and for all . If furthermore there exist a real number such that , then is said to be in the class . The union of taken over all possible sequences and all possible real numbers is denoted by .
Let denote the subclass of consisting of functions . Also Let denote the subclass of consisting of functions .

In this paper we obtain coefficient bounds for functions in the classes and , respectively, further we obtain distortion bounds and the extreme points for functions in these classes.

2. Coefficient Estimates

Unless otherwise mentioned, we assume in the reminder of this paper that , and .

We shall need the following lemmas.

Lemma 3. The sufficient condition for given by (1) to be in the class is that

Proof. It suffices to show that inequality (2) holds true. Upon using the fact that then inequality (2) may be written as or where and , then condition (2) or (9) is equivalent to We note that Using (11) and (12), then we can obtain the following inequality: The expression is bounded below by if or Hence the proof of Lemma 3 is completed.

By using (4) and (6) we can obtain the following lemma.

Lemma 4. A function of the form (1) is in the class if

Remark 5. Putting in Lemmas 3 and 4, we obtain the results obtained by Shams et al. [3, Theorems 2.1, 2.2, resp.].

In the following theorems, we show that the conditions (6) and (16) are also necessary for functions and , respectively.

Theorem 6. Let be of the form (1), then if and only if

Proof. In view of Lemma 3, we need only to show that function satisfies the coefficient inequality (17). If , then from (2), we have thus we have Since , lies in the class for some sequence and a real number such that Set in (19), then we obtain Letting , then we have Hence the proof of Theorem 6 is completed.

Corollary 7. If , then The equality holds for the function

Using the same technique used in Theorem 6 we get the following theorem

Theorem 8. Let of the form (1), then if and only if

Corollary 9. If , then The equality holds for the function

3. Distortion Theorems

Theorem 10. Let the function defined by (1) be in the class . Then The result is sharp.

Proof . We employ the same technique as used by Silverman [11]. In view of Theorem 6, since is an increasing function of , we have that is, Thus we have Similarly, we get This completes the proof of Theorem 10. Finally the result is sharp for the following function: at .

Corollary 11. Under the hypotheses of Theorem 8, is included in a disc with center at the origin and radius given by

Theorem 12. Let the function defined by (1) be in the class . Then The result is sharp.

Proof. Similarly is an increasing function of , in view of Theorem 6, we have that is, Thus we have Similarly Finally, we can see that the assertions of Theorem 12 are sharp for the function defined by (34). This completes the proof of Theorem 12.

Corollary 13. Under the hypotheses of Theorem 12, is included in a disc with center at the origin and radius given by

Using the same technique used in Theorems 10 and 12, we get the following theorems.

Theorem 14. Let the function defined by (1) be in the class . Then The result is sharp for the following function:

Theorem 15. Let the function defined by (1) be in the class . Then The result is sharp for the function given by (43).

4. Extreme Points

Theorem 16. Let the function defined by (1) be in the class , with , where . Define Then if and only if can be expressed in the form , where and .

Proof. If with and , then But according to (17), we can see that Then satisfies (17), hence .
Conversely, let the function defined by (1) be in the class , and define From Theorem 6, and so . Since , then This completes the proof of Theorem 16.

Finally using the same technique used in Theorem 16 we get the following theorem.

Theorem 17. Let the function defined by (1) be in the class , with , where (. Define Then if and only if can be expressed in the form , where and .

Acknowledgment

The authors thank the referees for their valuable suggestions which led to the improvement of this paper.

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