Sharp bounds for the ratio of q-gamma functions

Let Gamma (q) (0 < q <not equal> 1) be the q-gamma function and let s is an element of (0, 1) be a real number. We determine the largest number alpha = alpha (q,s) and the smallest number beta = beta (q,s) such that the inequ alities (1 - q(x+alpha)/1 - q)(1-s) < <Gamma>(q)(x + 1)/Gamma (q)(x + s) < (1 - q(x +<beta>)/1 - q)(1-s) hold for all positive real numbers rr. Our result refines and extends recen tly published inequalities by ISMAIL and MULDOON (1994).