函数[Γ(x)]^(1/x)·e^(α/x)的几何凸性及其应用

The Geometrically Convexity of [Γ(x)]^(1/x)·e^(α/x) and Its Application

设Γ为(0,+∝)上的Gamma函数,Ψ(x)=ΓΓ′((xx))和fα(x)=[Γ(x)]1x.exα,α∈R,x∈(0,+∝),本文研究了函数fα的几何凸性,得到一个关于Gamma函数且含参数的不等式,同时证明了:当n∈N,n 1时,有n+11-ln2nn+2(n!)n+111 n+11+ln2nn-2成立,其加强了Minc-Sather不等式。

Let Г be Gamma function on （0,＋∝）, and ψ（x）=Г′（x）/Г（x）和fa（x）=[Г（x）]^1/x·e^a/x,a∈R,x∈（0,＋∝）. We present the Geometrically Convexity of fa in this paper. As the application, we prove （（n＋1）/n）^[1-（1nn＋2）/2n]≤（n!）^1/（n＋1）/（（n-1）!）^1/n≤（（n＋1）/n）^[1＋（1nn-2）/2n] with n∈N and n≥1. It sharpens Minc Sather Inequality.