Bounds for radii of convexity of some $q$-Bessel functions

TL;DR

This paper derives bounds for the radii of convexity of certain $q$-Bessel functions using normalization techniques, inequalities, and properties of entire functions, providing solutions to transcendental equations.

Contribution

It introduces new bounds for the convexity radii of $q$-Bessel functions through innovative normalization and analytical methods, including Euler-Rayleigh inequalities.

Findings

Derived tight bounds for convexity radii

Identified solutions as roots of transcendental equations

Applied Laguerre-Pólya class properties

Abstract

In the present investigation, by applying two different normalizations of the Jackson and Hahn-Exton $q$-Bessel functions tight lower and upper bounds for the radii of convexity of the same functions are obtained. In addition, it was shown that these radii obtained are solutions of some transcendental equations. The known Euler-Rayleigh inequalities are intensively used in the proof of main results. Also, the Laguerre-P\'olya class of real entire functions plays an important role in this work.