Abstract Let S denote the class of functions f analytic and univalent in the unit disk | z | 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 ∈ S , then F is univalent for | z | 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 ∈ A satisfying the functional inequality | f ′ ( z ) ( z f ( z ) ) 2 − 1 | 1 , ∣ z ∣ 1 . Some other related results are also presented.