Starlikeness of a cross-product of Bessel functions

Abstract

In this paper a necessary and sufficient condition is deduced for the close-to-convexity of a cross-product of Bessel and modified Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives, the newly discovered power series and infinite product representation of this crossproduct, as well as a slightly modified version of a result of Lorch on the monotonicity of the zeros of the cross-product with respect to the order.