The radius of univalence of the reciprocal of a product of two analytic functions

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Let A denote the family of all functions f analytic in the open unit disk with the normalization f(0)=0= f'(0)-1 and S be the class of univalent functions from A. In this paper, we consider radius of univalence of F defined by F(z)=z³/(f(z)g(z)), where f and g belong to some subclasses of A (for which f(z)/z and g(z)/z are non-vanishing in) and, in some cases in precise form, belonging to some subclasses of S. All the results are proved to be sharp. Applications of our investigation through Bessel functions are also presented.