与Gamma函数有关的对数完全单调函数及其应用

摘要

In order to improve the logarithmically completely monotonicity and related inequalities of the function Gα,β(x)(where α∈R, β ≥0 are parameter) and 1/Gα,β(x) which are defined in (0,∞). Using Taylor series expansion, the series expansion and integral expression of Gamma function and Psi function, this paper researches the logarithmically completely monotonicity of function Gα,β(x) and 1/Gα,β(x) and expands the sufficient condition. By the logarithmically completely monotonicities, a new inequality is established. Based on the research of the special circumstances, a symmetrical and concise two-side inequality, which estimates the division of factorial and∏nk=1 kk, is established.%为了完善函数 Gα,β(x)(其中参数α∈ R,β≥0)及函数1/Gα,β(x)在区间(0,∞)上的对数完全单调性和相关不等式,利用Taylor展开式、Gamma函数、Psi函数的级数表达式和积分表达式研究了函数 Gα,β(x)和函数1/Gα,β(x)数的对数完全单调性,将函数Gα,β(x)和函数1/Gα,β(x)对数完全单调的充分条件扩大；利用对数完全单调性得到新的不等式,并通过对特殊情形的研究,得到一个形式简单对称的双边不等式,该不等式对阶乘数之乘积与∏nk=1 kk 的商做出估计。