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