Sharp bounds for the q-gamma function in terms of the Lambert W function

Abstract

In this paper, the function $$F_a(x;q)$$ is defined in terms of logarithmic of q-gamma function for all reals x, a and q with $$q>0$$ . The values of a in which the function $$F_a(x;q)$$ is completely monotonic function are determined and it turns out that these values depend on the Lambert W function. As a consequence, sharp upper and lower bounds for the q-gamma, q-digamma and q-polygamma functions for all positive real q in terms of the Lambert W function are provided. Analytically and numerically, our results are compared with previous results. Furthermore, best bounds for the Lambert W function are provided.