Complete characterizations of the gamma function

In this paper, we obtain the complete monotonicities regarding both the functions σ x and λ x defined by Γ ( x + 1 ) = 2 π x x e x e σ x 12 x = 2 π x x e x ( 1 + λ x ) . As direct consequences, σ x is strictly increasing and strictly concave on ( 0 , ∞ ) and λ x is strictly decreasing and strictly convex on ( 0 , ∞ ) . Furthermore, we show that σ x consists the properties lim x → 0 + σ x = 0 and lim x → ∞ σ x = 1 as well as λ x holds the properties lim x → 0 + λ x = ∞ and lim x → ∞ λ x = 0 .