On Some Classes of Multivalent Starlike Functions

Classes of multivalent functions analogous to certain classes of univalent starlike functions are defined and studied. Estimates on coefficients and distortion are made, using a variety of techniques. 1. Let St denote the class of all functions/(z) = z + . . . analytic, univalent and starlike in the unit disc U. Such functions satisfy the condition Re(z/'(z)//(z)) >0,zGU. The problem of defining a corresponding class of multivalent starlike functions has been studied by several authors. Hummel [5] distinguishes six commonly used definitions, a typical one being/(z) belongs to the class S(p) if/has at most p zeros in U and zf (z) (I i) lim sup min Re -, , > 0. y ' r-+\V \z\=r f(z) In this note we will study three classes of multivalent starlike functions which are analogues of certain subclasses of St. 2. Let Sx(p, a), p a positive integer, 0 < a < 2p, denote the class of all functions/(z) = a0 + ajZ + . . . analytic in <7 with precisely p zeros there such that zf'(z) (2.1) lim sup min Re —rr— > a. V r-l1^ \z\ = r f(z) Sx (p, a) is the generalization of the class S(a) of starlike functions of order a introduced by Robertson [11]. Theorem 2.1. Let f(z) belong to the class Sx(p, a) and suppose f has zeros at zx, z2.z . 77zen f(z) is p-valent in U and there is a function g in the class S(a/p) and a constant A such that (2.2) /(z) = ¿n*(z,z,)fe(z)jp, /=i Received by the editors May 6, 1974. AMS (MOS) subject classifications (1970). Primary 30A32, 30A34.