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

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 , .


Introduction
Let A denote the class of all functions  that are analytic in the open unit disk  = { : || < 1} and are normalized by the conditions (0) =   (0) − 1 = 0. We denote by S the subclass of A consisting of functions which are univalent in  and denote by S * (), 0 ≤  < 1, the class of functions in S which are starlike of order  in .Analytically, S * () = { ∈ A : Re (   ()  () ) > ,  ∈ } .
Recently, Obradovic and Ponnusamy [9] investigated another class where  > 0 and ()/ ̸ = 0 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 M().The purpose of the present paper is to answer this question.where ()/ ̸ = 0 in ,  > 0, and  ∈ R \ [−1/2, 0].We find conditions on , , and  involved in the class M(, ) under which members of M(, ) are starlike of a given order , 0 ≤  < 1.We remark that the class M(, ) follows essentially from the class (, , ) studied by Baricz and Ponnusamy [10] by taking  = 0; however, we will study those issues for the class M(, ) which are not studied by the authors for (, , ).

Main Results
Let A  denote the class of analytic functions  in  such that  () (0) = 0 for  = 0, 1, 2, . . ., , where We begin with the following result.
Proof.As  ∈ M(, ), there exists a Schwarz function  ∈ B 1 such that Since  ∈ B 1 , (0) =   (0) = 0.If we set where  is an analytic function in  with (0) =   (0) = 0, then ( 5) is equivalent to from which we get or, equivalently, Solving this equation for /(), we obtain In the next result, we find the range of values of  for which  ∈ M(, ) implies that  ∈ (), 0 <  ≤ 1.
In the following theorem, we find conditions under which functions in the class M(, ) belong to S * (), 0 ≤  < 1.
A simple calculation gives Taking  = 0, in Theorem 5, we get the following result.