Geometric perspective of generalized Bessel function

Abstract In this study, we extend and refine several results concerning the geometric properties of generalized Bessel functions established by Á. Baricz (Mathematica 48(71):1318, 2006). The analysis focuses on cases where the parameters lie within a bounded domain. The primary methodology employs classical sufficient conditions for univalency, convexity, starlikeness, and close-to-convexity of analytic functions defined within the open unit disk, as originally formulated by Ozaki (Sci. Rep. Tokyo Bunrika Daigaku 2:167–188, 1935) and Mocanu (Libertas Math. 13:27–40, 1993). Additionally, this work explores the uniform convexity and starlikeness of the normalized form of the Bessel function. To demonstrate the validity and applicability of the theoretical framework, several specific examples are provided.