Geometric Properties of Bessel function derivatives

TL;DR

This paper investigates the geometric properties of derivatives of Bessel functions, specifically their radii of starlikeness and convexity, using Mittag-Leffler expansions and zero properties, extending known results.

Contribution

It introduces new bounds and characterizations for the radii of starlikeness and convexity of derivatives of Bessel functions with three different normalizations.

Findings

Derived tight bounds for radii of starlikeness and convexity.

Extended classical results to derivatives of Bessel functions.

Utilized Mittag-Leffler expansion and zero properties in proofs.

Abstract

In this paper our aim is to find the radii of starlikeness and convexity of Bessel function derivatives for three different kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for nth derivative of Bessel function and properties of real zeros of it. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized nth derivative of Bessel function. The main results of the paper are natural extensions of some known results on classical Bessel functions of the first kind.