Asymptotic Expansions for the Radii of Starlikeness of Normalized q-Bessel Functions

Abstract

In this paper we study the asymptotic behavior of the radii of starlikeness of the normalized Jackson’s second and third q-Bessel functions, focusing on their large orders. To achieve this, we use the Rayleigh sums of positive zeros of both q-Bessel functions, and determine the coefficients of the asymptotic expansions. We derive complete asymptotic expansions for these radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums of positive zeros of q-Bessel functions, Kvitsinsky’s results on spectral zeta functions for q-Bessel functions and asymptotic inversion. Moreover, we derive bounds for the radius of starlikeness of q-Bessel functions by using potential polynomials.