Geometric Properties of Generalized Bessel Function associated with the Exponential Function

TL;DR

This paper investigates the geometric properties of generalized Bessel functions, establishing conditions for their starlikeness and convexity within the unit disk, and deriving related differential subordination results with illustrative examples.

Contribution

It provides new sufficient conditions for generalized Bessel functions to belong to subclasses of starlike and convex functions, extending existing differential subordination frameworks.

Findings

Conditions for starlikeness and convexity of Bessel functions

Differential subordination implications involving Bessel functions

Examples with trigonometric and hyperbolic functions

Abstract

Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz \emph{et al.} [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. {\bf 38} (2015), no.~3, 1255--1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results.