The radius of convexity of normalized Bessel functions of the first kind

TL;DR

This paper determines the radius of convexity for normalized Bessel functions of the first kind, establishing their geometric properties and disproving a previous conjecture about their convexity.

Contribution

It introduces new Mittag-Leffler expansions and analyzes zeros of Bessel functions to find convexity radii and optimal parameters, also disproving a conjecture.

Findings

Normalized Bessel functions are convex-univalent on certain disks.

The paper finds the exact radius of convexity for these functions.

It disproves a conjecture regarding the convexity of Bessel functions.

Abstract

In this paper we determine the radius of convexity for three kind of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. Moreover, we find the optimal parameters for which these normalized Bessel functions are convex in the open unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy concerning the convexity of the Bessel function of the first kind.