Abstract
In the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions Gamma q(x+1)/Gamma q(x+s) for all real number s and 0<q<not equal>1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309-323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5-14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all 0<s<1 when letting q -> 1 and a new inequality for all s>1.