Inequalities for the Gamma Function

We prove the following two theorems: (i) Let Mr(a, b) be the rth power mean of a and b. The inequality Mr(Γ(x), Γ(1/x)) ≥ 1 holds for all x ∈ (0,∞) if and only if r ≥ 1/C − π2/(6C2), where C denotes Euler’s constant. This refines results established by W. Gautschi (1974) and the author (1997). (ii) The inequalities xα(x−1)−C < Γ(x) < xβ(x−1)−C (∗) are valid for all x ∈ (0, 1) if and only if α ≤ 1−C and β ≥ (π2/6−C)/2, while (∗) holds for all x ∈ (1,∞) if and only if α ≤ (π2/6− C)/2 and β ≥ 1. These bounds for Γ(x) improve those given by G. D. Anderson an S.-L. Qiu (1997).