Radii of Starlikeness and Convexity of a Product and Cross-Product of Bessel Functions

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are considered and for each of the six functions tight lower and upper bounds are given for the radii of starlikeness and convexity, via Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that four from the six normalized functions are convex in the open unit disk.