Product of univalent functions
Let S denote the class of functions f analytic and univalent in the unit disk vertical bar z vertical bar < 1 normalized such that f (0) = 0 = f'(0) - 1. In this article the authors discuss the radius of univalence of F (z) = g(z) h(z)/z when g and h belong to certain subsets of S. The paper concludes with the following conjecture. If g, h is an element of S, then F is univalent for vertical bar z vertical bar < 1/3 and the number 1/3 cannot improved. The conjecture is shown to be true for some subclasses of S, e.g. the class of starlike functions, and the class U consisting of functions f is an element of A satisfying the functional inequality vertical bar f'(z) (z/f(z))(2) - 1 vertical bar < 1, vertical bar z vertical bar < 1. Some other related results are also presented. (C) 2012 Elsevier Ltd. All rights reserved.
univalent analytic radius of univalency starlike functions coefficient inequality area theorem