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We prove among others results that the harmonic mean of ?q(x) and ?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 p0 ? (1, 9/2), such that for q ? (0, p0), ?q(1) is the minimum of the harmonic mean of ?q(x) and q(1/x) for x > 0 and for q ? (p0,+?), ?q(1) is the maximum. Our results generalize some known inequalities due to Alzer and Gautschi.doi:10.2298/fil2112105b fatcat:5zwmy5apxnadzoqbcbw5qbswqm