Radii of starlikeness and convexity of a cross-product of Bessel functions

In this paper, 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 normalizations are investigated and for each of the six functions, the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike in the open unit disk; however, the convex case is open for further research. The characterization of entire functions from the Laguerre–Pólya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing properties of the zeros of the cross-product and product of Bessel functions and their derivatives are also useful in the proof of the main results. © 2017, Springer Science+Business Media, LLC.