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

Abstract

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. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. 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 investigated and for each of the six functions the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. For these radii of starlikeness and convexity tight lower and upper bounds are given via Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike and convex in the open unit disk. The properties and the characterization of real entire functions from the Laguerre-P\'{o}lya class via hyperbolic polynomials play an important role in this paper. Some open problems are also stated, which may be of interest for further research.