Bounds for the radii of univalence of some special functions

TL;DR

This paper establishes precise bounds for the univalence radii of certain special functions, revealing relationships between different classes and employing advanced inequalities and function classes.

Contribution

It provides the first tight bounds for the univalence radii of normalized Bessel, Struve, and Lommel functions, and links univalence to starlikeness using Laguerre-Pólya class insights.

Findings

Bounds for univalence radii of Bessel, Struve, Lommel functions

Struve functions have larger univalence radii than Bessel functions

Univalence radii coincide with starlikeness radii for certain functions

Abstract

Tight lower and upper bounds for the radius of univalence of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, by using the idea of Kreyszig and Todd, and Wilf it is proved that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. The Laguerre-P\'olya class of entire functions plays an important role in our study.