On some geometric properties and Hardy class of q-Bessel functions

TL;DR

This paper investigates geometric properties like starlikeness and convexity of q-Bessel functions, extending classical Bessel functions, and explores conditions under which these functions belong to Hardy and bounded analytic classes.

Contribution

It provides new results on the geometric behavior and Hardy class membership of Jackson's second and third q-Bessel functions, extending classical Bessel function theory.

Findings

Conditions for starlikeness and convexity of q-Bessel functions.

Criteria for q-Bessel functions to belong to Hardy space.

Criteria for q-Bessel functions to be bounded analytic.

Abstract

In this paper, we deal with some geometric properties including starlikeness and convexity of order $\alpha$ of Jackson's second and third $q$-Bessel functions which are natural extensions of classical Bessel function $J_{\nu}$. In additon, we determine some conditions on the parameters such that Jackson's second and third $q$-Bessel functions belong to the Hardy space and to the class of bounded analytic functions.