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

Let $\mathcal{A}$ denote the family of all functions $f$ analytic in the open unit disk $\text{ID}$ with the normalization $f(0)=0=f^{\mathrm{\prime }}(0)-1$ and ${\mathcal S}$betheclassofunivalentfunctionsfrom${\mathcal A}$. In this paper, we consider radius of univalence of $F$ defined by $F(z)=z^3\mathrm{/}(f(z)g(z))$, where $f$ and $g$ belong to some subclasses of $\mathcal{A}$ (for which $f(z)\mathrm{/}z$ and $g(z)\mathrm{/}z$ are non-vanishing in $\text{ID}$) and, in some cases in precise form, belonging to some subclasses of $\mathcal{S}$. All the results are proved to be sharp. Applications of our investigation through Bessel functions are also presented.