Univalence and starlikeness of certain transforms defined by convolution of analytic functions

摘要

Let U(lambda) denote the class of all analytic functions f in the unit disk Delta of the form f (z) = z + a(2)z(2) + ... satisfying the condition f(z)(z/f(z))(2) -1 vertical bar <= lambda, z is an element of Delta. In this paper we find conditions on lambda and on c is an element of C with Rec >= 0 not equal c such that for each f is an element of u(lambda) satisfying (z/f (z))*F(1, c; c + 1; z) not equal 0 for all z is an element of Delta the transform G(z) = G(f)(c) (z) = z/(z/f(z))*F(1, c; c + 1; z), z is an element of Delta, is univalent or starlike. Here F(a, b; c; z) denotes the Gauss hypergeometric function and * denotes the convolution (or Hadamard product) of analytic functions on Delta. (c) 2007 Elsevier Inc. All rights reserved.