The radius of $\alpha$-convexity of normalized Bessel functions of the first kind

TL;DR

This paper determines the radii of lpha-convexity for normalized Bessel functions of the first kind, showing they lie between starlikeness and convexity radii and decrease with lpha, unifying previous results.

Contribution

It introduces a unified approach to compute lpha-convexity radii for Bessel functions, extending and connecting existing results on starlikeness and convexity.

Findings

Radii of lpha-convexity are between starlikeness and convexity radii.

These radii decrease as lpha increases.

Results unify previous findings on Bessel function geometries.

Abstract

The radii of $\alpha$-convexity are deduced for three different kind of normalized Bessel functions of the first kind and it is shown that these radii are between the radii of starlikeness and convexity, when $\alpha\in[0,1],$ and they are decreasing with respect to the parameter $\alpha.$ The results presented in this paper unify some recent results on the radii of starlikeness and convexity for normalized Bessel functions of the first kind. The key tools in the proofs are some interlacing properties of the zeros of some Dini functions and the zeros of Bessel functions of the first kind.