Product of univalent functions

Let SS denote the class of functions ff analytic and univalent in the unit disk |z|<1|z|<1 normalized such that f(0)=0=f′(0)−1f(0)=0=f′(0)−1. In this article the authors discuss the radius of univalence of F(z)=g(z)h(z)/zF(z)=g(z)h(z)/z when gg and hh belong to certain subsets of SS. The paper concludes with the following conjecture. If g,h∈Sg,h∈S, then FF is univalent for |z|<1/3|z|<1/3 and the number 1/31/3 cannot improved. The conjecture is shown to be true for some subclasses of SS, e.g. the class of starlike functions, and the class UU consisting of functions f∈Af∈A satisfying the functional inequality |f′(z)(zf(z))2−1|<1,∣z∣<1. Some other related results are also presented.