Complete characterizations of the gamma function

In this paper, we obtain the complete monotonicities regarding both the functions σxσx and λxλx defined by Γ(x+1)=2πxxexeσx12x=2πxxex(1+λx). As direct consequences, σxσx is strictly increasing and strictly concave on (0,∞)(0,∞) and λxλx is strictly decreasing and strictly convex on (0,∞)(0,∞). Furthermore, we show that σxσx consists the properties limx→0+σx=0limx→0+σx=0 and limx→∞σx=1limx→∞σx=1 as well as λxλx holds the properties limx→0+λx=∞limx→0+λx=∞ and limx→∞λx=0limx→∞λx=0.