Starlikeness of a cross-product of Bessel functions

TL;DR

This paper establishes a precise criterion for the close-to-convexity of a cross product of Bessel functions, expanding understanding of their geometric properties using advanced complex analysis techniques.

Contribution

It provides the first necessary and sufficient condition for the close-to-convexity of this specific cross product of Bessel functions and their derivatives.

Findings

Derived a new power series and infinite product representation.

Established a criterion for close-to-convexity based on zeros analysis.

Extended existing results on Bessel functions' geometric properties.

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 cross-product, 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.