Geometric and monotonic properties of hyper-Bessel functions

TL;DR

This paper investigates geometric properties of normalized hyper-Bessel functions, focusing on radii of starlikeness, convexity, and bounds for zeros, using advanced inequalities and function classes.

Contribution

It provides new results on the radii of geometric properties and bounds for hyper-Bessel functions, employing Euler-Rayleigh inequalities and Laguerre-Pólya class techniques.

Findings

Radii of starlikeness, convexity, and uniform convexity are characterized.

Transcendental equations for the radii are derived.

Bounds for the first positive zero of hyper-Bessel functions are established.

Abstract

Some geometric properties of a normalized hyper-Bessel functions are investigated. Especially we focus on the radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions and we show that the obtained radii satisfy some transcendental equations. In addition, we give some bounds for the first positive zero of normalized hyper-Bessel functions, Redheffer-type inequalities, and bounds for this function. In this study we take advantage of Euler-Rayleigh inequalities and Laguerre-P\'{o}lya class of real entire functions, intensively.