Complete characterizations of the gamma function

摘要

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