Geometric Properties of Functions Containing Derivatives of Bessel Functions

TL;DR

This paper investigates the geometric properties of functions involving derivatives of Bessel functions, specifically focusing on radii of various spirallike and convex classes, and explores their starlike and convex radii with visual verification.

Contribution

It introduces new results on the radii of gamma-spirallike and convex gamma-spirallike functions for normalized Bessel-related functions, including visual and tabular analysis.

Findings

Derived radii for gamma-spirallike functions

Computed convex gamma-spirallike radii

Provided visual verification with graphs

Abstract

In this paper our aim is to find the radii of $\gamma$-Spirallike of order $\alpha$ and convex $\gamma$-Spirallike of order $\alpha$ for three different kinds of normalizations of the function $N_\nu(z)=az^2J_\nu^{\prime\prime}(z)+bzJ_\nu^{\prime}(z)+cJ_\nu(z),$ where $J_\nu(z)$ is the Bessel function of the first kind of order $\nu.$ Moreover, the $\mathcal{S}^*\left(\varphi\right)-$radii and $\mathcal{C}\left(\varphi\right)-$radii of these normalized functions are investigated. The tables are created and visual verification with graphs are made by giving special values to the real numbers $a,$ $b$ and $c$ in the obtained results.