Some Characterizations on the Normalized Lommel, Struve and Bessel Functions of the First Kind

TL;DR

This paper develops new techniques to establish necessary and sufficient conditions for normalized Lommel, Struve, and Bessel functions of the first kind to belong to subclasses of starlike and convex functions, enhancing understanding of their geometric properties.

Contribution

It introduces novel methods for characterizing the geometric properties of these special functions within the framework of univalent function theory.

Findings

Derived conditions for starlikeness and convexity of the functions.

Extended the class of functions known to have specific geometric properties.

Provided a unified approach for multiple special functions.

Abstract

In this paper, we introduce new technique for determining some necessary and sufficient conditions of the normalized Bessel functions $j_{\nu}$, normalized Struve functions $h_{\nu}$ and normalized Lommel functions $s_{\mu,\nu}$ of the first kind, to be in the subclasses of starlike and convex functions of order $\alpha$ and type $\beta$.