International Journal of Analysis

International Journal of Analysis / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 185635 | 4 pages |

Geometric Properties of a Class of Analytic Functions Defined by a Differential Inequality

Academic Editor: Baruch Cahlon
Received08 Jul 2015
Revised10 Sep 2015
Accepted15 Sep 2015
Published04 Oct 2015


Let be the class of analytic functions defined in the open unit disk and normalized by . For in , let , where and . In the present paper, we find conditions under which functions in the class are starlike of order , .

1. Introduction

Let denote the class of all functions that are analytic in the open unit disk and are normalized by the conditions . We denote by the subclass of consisting of functions which are univalent in and denote by , , the class of functions in which are starlike of order in . Analytically,Note that is the usual class of starlike (with respect to the origin) functions in and we denote it simply by . We say that , , if and , for all . It is well known that the functions in are close-to-convex and hence univalent in [1, 2].

In 1972, Ozaki and Nunokawa [3] studied the class , which is defined aswhere in . They proved that for . Several researchers studied this class (e.g., see [48]) and obtained many significant results.

Recently, Obradovic and Ponnusamy [9] investigated another class where and in . They obtained certain inclusion relations, characterization formula, and coefficient conditions. They also posed a question about the starlikeness of the functions in the class . The purpose of the present paper is to answer this question. In fact, we study a more general class where in , , and . We find conditions on , , and involved in the class under which members of are starlike of a given order , . We remark that the class follows essentially from the class studied by Baricz and Ponnusamy [10] by taking ; however, we will study those issues for the class which are not studied by the authors for .

2. Main Results

Let denote the class of analytic functions in such that for , where . With , we set : is analytic, in and for . Functions in are called Schwarz functions. Obviously, implies that in , for .

We begin with the following result.

Lemma 1. Let be in . Then, in whenever .

Proof. As , there exists a Schwarz function such that Since , . If we setwhere is an analytic function in with , then (5) is equivalent tofrom which we getor, equivalently,Solving this equation for , we obtainNow, using the fact that , it follows thatThe inequality (11) is equivalent to which givesfor all whenever .

In the next result, we find the range of values of for which implies that , .

Theorem 2. Let be in . Then, for , where satisfies the inequality

Proof. From (8), we getwhereTherefore,Also, in view of (11), we obtainTherefore,Now, the desired result follows, ifBy using , and and carrying out some simplifications, we conclude that (20) is equivalent to (14).

If and , then Theorem 2 gives the following.

Corollary 3. If is in , and then in , whenever .

Setting , , and in Theorem 2, we obtain the following.

Corollary 4. If is in , and then in , whenever .

In the following theorem, we find conditions under which functions in the class belong to , .

Theorem 5. Let be in and let . Then, for , provided .

Proof. As , from (8) and (10), we getNow, is equivalent toUsing (21) in (22), we getor, equivalently,If we denote the left-hand side of (24) by and let then, in view of the rotation invariance property of the set , we obtain that (22) holds if .
A simple calculation gives Now, by the parallelogram Law, , we have Using the fact that in , we getThus, , if .

Taking , in Theorem 5, we get the following result.

Corollary 6. Let be as in Theorem 5. Then, provided .

Conflict of Interests

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


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Copyright © 2015 Manpreet Kaur 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.

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