Monotonicity, convexity, and complete monotonicity of two functions related to the gamma function

被引：12
|
作者：
Yang, Zhen-Hang [1 ,2 ]
Tian, Jing-Feng [3 ]
机构：
[1] North China Elect Power Univ, Coll Sci & Technol, Ruixiang St 282, Baoding 071051, Peoples R China
[2] State Grid Zhejiang Elect Power Co Res Inst, Dept Sci & Technol, Hangzhou 310014, Zhejiang, Peoples R China
[3] North China Elect Power Univ, Dept Math & Phys, Yonghua St 619, Baoding 071003, Peoples R China
来源：
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS | 2019年 / 113卷 / 04期
关键词：
Gamma function; Monotonicity; Convexity; Complete monotonicity; More accurate bounds; ASYMPTOTIC EXPANSIONS; APPROXIMATION; INEQUALITIES; FORMULAS; SERIES;
D O I：
10.1007/s13398-019-00719-z
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
In this paper, we prove that for a = 31/ 98, the function fa (x) = ln x + 1 2 -x ln x + x -1 2 ln (2p) + x 24 x2 + a -7/120 x4 + ax2 + (98a -31) /1680 is strictly increasing (decreasing) and concave (convex) on (0,8) if and only if a = 5281/ 6068 (a = 31/ 98). Moreover, we show that the necessary and sufficient condition for function Fa (x) = - x4 + ax2 + 98a -31 1680 fa (x) for a. R to be completely monotonic on ( 0,8) is also a = 5281/ 6068. These yield some new sharp bounds for the gamma function.
引用
收藏
页码：3603 / 3617
页数：15
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