Sharp Bounds for the Ratio of q – Gamma Functions

Let Γ q (0 < q ≠ 1) be the q –gamma function and let s ∈ (0, 1) be a real number. We determine the largest number α = α ( q , s ) and the smallest number β = β ( q , s ) such that the inequalities$$\left ( {{1 - q^{x+\alpha}} \over {1 - q}} \right ) ^{1 - s} < {{\Gamma _q (x + 1)} \over {\Gamma _q (x + s)}} < \left ( {{1 - q ^{x+\beta}} \over {1 - q}} \right ) ^{1 - s}$$hold for all positive real numbers x . Our result refines and extends recently published inequalities by Ismail and Muldoon (1994).