Univalence of certain integral operators involving q-Bessel functions

Abstract

In this investigation we make a systematic study of the univalence of certain families of integral operators, which are defined by means of the normalized forms of Jackson's second and third q-Bessel functions. where Γ stands for the Euler gamma function, z ∈ C and ν ∈ R. In his research of basic numbers, F.H. Jackson [1-3] defines the oldest q-analogues of the Bessel functions, namely J (1) ν (z; q). In the following years, the Jackson's second and third q-Bessel functions [4, 5] were defined by: J (2) ν (z; q) = (q ν+1 ; q) ∞ (q; q) ∞ n≥0 (−1) n z 2 2n+ν (q; q) n (q ν+1 ; q) n q n(n+ν) , (2) and J (3) ν (z; q) = (q ν+1 ; q) ∞ (q; q) ∞ n≥0 (−1) n z 2n+ν (q; q) n (q ν+1 ; q) n q 1 2 n(n+1) , (3) where z ∈ C, ν > −1, q ∈ (0, 1) and (a; q) 0 = 1, (a; q) n = n k=1 (1 − aq k−1), (a; q) ∞ = k≥1 (1 − aq k−1).