Supplements to Known Monotonicity Results and Inequalities for the Gamma and Incomplete Gamma Functions

In particular they proved that for x > 0 and α = 0 the function [Γ(1 + 1/x)]x decreases with x, while when α=1 the function x[Γ(1+1/x)]x increases.Moreover they also showed that the values α= 0 and α= 1, in the properties mentioned above, cannot be improved if x ∈ (0,+∞). In this paper we continue the investigation on the monotonicity properties for the gamma function proving, in Section 2, the following theorem. Theorem 1.1. The functions f (x) = Γ(x + 1/x), g(x) = [Γ(x + 1/x)]x and h(x) = Γ′(x + 1/x) decrease for 0 < x < 1, while increase for x > 1.