On some geometric properties and Hardy class of &lt;i&gt;q&lt;/i&gt;-Bessel functions

,İbrahim Aktaş

doi:10.3934/math.2020203

Abstract

In this paper, we deal with some geometric properties including starlikeness and convexity of order α of Jackson's second and third q-Bessel functions which are natural extensions of classical Bessel function Jν. 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.

Highlights

Introduction and PreliminariesSpecial functions appears in many branches of mathematics and applied sciences
The Bessel function of the first kind Jν is a particular solution of the following homogeneous differential equation: z2w′′(z) + zw′(z) + z2 − ν2 w(z) = 0, which is known as Bessel differential equation
We present some results regarding Hardy space of normalized q-Bessel functions

Summary

Introduction and Preliminaries

Special functions appears in many branches of mathematics and applied sciences. One of the most important special functions is Bessel function of the first kind Jν. Convex and starlike functions of order α, Hardy space, q-Bessel functions. Let Hp (0 < p ≤ ∞) denote the Hardy space of all analytic functions f (z) in U, and define the integral means Mp(r, f ) by (1.8). The authors in [7, 16, 17, 22, 23, 28, 29] studied the Hardy space of some special functions (like Hypergeometric, Bessel, Struve, Lommel and Mittag-Leffler) and analytic function families. We find some conditions for the hadamard products h(ν2)(z; q) ∗ f (z) and h(ν3)(z; q) ∗ f (z) to belong to H∞ ∩ R, where h(νk)(z; q) are Jackson’s normalized q-Bessel functions which are given by (2.1) and (2.2) for k ∈ {2, 3}, and f is an analytic function in R. 1 2 for some k, l ∈ C and θ ∈ R, the following statements hold: a

If α

If If

Hp for p

This means function z