Convex and starlike univalent functions

Abstract

Introduction. Let (S) denote the class of functions/(z) = z + 2? anzn which are regular and univalent in \z\ < 1 and which map \z\ < 1 onto domains D(f).Let (C), (S*), and (K) represent the subclasses of (S) where D(f) are respectively, close-to-convex, starlike with respect to the origin, and convex.It follows that (K)<=(S*)<=(C)<=(S).We will simply say "starlike" when we mean starlike with respect to the origin, and the statement "f(z) is convex" will mean that the domain D(f) is convex.The abbreviations "i.o.i." and "n.s.c." have the usual meanings.Let (F) denote the class of functions p(z) which are regular and satisfy p(0)=l, Rep(z)>0 for \z\ < 1.The following results, which we will use repeatedly, are well known [1]:Let/(z)e(S).ThenIf h(z) = z+ -is regular in |z| < 1 and h'(z) e (P) then h(z) e (S).In a recent paper [4] R. J. Libera established that (1.1) iff(z) is a member of(C), (S*) or (K) then F(z) = (2/z)Jlf(t) dt is also a member of the same class, respectively.The purpose of this paper is to construct a many-parameter class of functions F(c;/) which includes the result (1.1) for special choice of parameters c and, in addition, establish a subordinate relation between the image domains of certain members of F(ct;f) and the image domain of the parent function/(z).2. Definitions and lemmas.The notation of g(z)<f(z), ("g(z) is subordinate to/(z)"), will mean that every value taken by g(z) for \z\ < 1 is also taken by/(z).The convolution of two power series/=2fAnzn and g=2 Bnzn is defined as the power series/* g=2i AnBnzn.Definition 1.An infinite sequence {bn}x of complex numbers is called a subordinating factor sequence (s.f.s.) if whenever f(z) e (K) we have E(z)=f(z) * 2f ^nZn<f(z).This definition is due to H. S. Wilf [11] who gave n.s.c. for a sequence to be a s.f.s.It follows that the product sequence {bncn}x of two s.f.s.{>"}f and {cn}x is also a s.f.s.