On circularly symmetric functions

Abstract

Let D C and 02 D. A set D is circularly symmetric if for each % 2 R + a set D\f 2 C : j j = %g is one of three forms: an empty set, a whole circle, a curve symmetric with respect to the real axis containing %. A function f 2 A is circularly symmetric if f() is a circularly symmetric set. The class of all such functions we denote by X. The above denitions were given by Jenkins in (2). In this paper besides X we also consider some of its subclasses: X( ) and Y \S consisting of functions in X with the second coecient xed and univalent starlike functions respectively. According to the suggestion, in Abstract we add one more paragraph at the end of the section: For X( ) we nd the radii of starlikeness, starlikeness of order , univalence and local univalence. We also obtain some distortion results. For Y\S we discuss some coecient problems, among others the Fekete- Szego ineqalities.