A Harmonic Mean Inequality for the q-Gamma and q-Digamma Functions

We prove among others results that the harmonic mean of Gamma(q)(x) and Gamma(q)(1/x) is greater than or equal to 1 for arbitrary x > 0, and q is an element of J where J is a subset of [0, +infinity). Also, we prove that there is a unique real number p(0) is an element of (1, 9/2), such that for q is an element of (0, p(0)), psi(q)(1) is the minimum of the harmonic mean of psi(q)(x) and psi(q)(1/x) for x > 0 and for q is an element of (p(0), +infinity), psi(q)(1) is the maximum. Our results generalize some known inequalities due to Alzer and Gautschi.