Abstract
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 [4–8]) 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.