Monotonicity and inequalities involving the incomplete gamma function

被引：0
作者：
Zhen-Hang Yang
Wen Zhang
Yu-Ming Chu
机构：
[1] Hunan City University,School of Mathematics and Computation Sciences
[2] State Grid Zhejiang Electric Power Research Institute,Customer Service Center
[3] Yeshiva University,Albert Einstein College of Medicine
来源：
Journal of Inequalities and Applications | / 2016卷
关键词：
incomplete gamma function; gamma function; psi function; 33B20; 26D07; 26D15;
D O I：
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摘要：
In the article, we deal with the monotonicity of the function x→[(xp+a)1/p−x]/Ip(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\rightarrow[ (x^{p}+a )^{1/p}-x]/I_{p}(x)$\end{document} on the interval (0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0, \infty)$\end{document} for p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p>1$\end{document} and a>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a>0$\end{document}, and present the necessary and sufficient condition such that the double inequality [(xp+a)1/p−x]/a<Ip(x)<[(xp+b)1/p−x]/b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[ (x^{p}+a )^{1/p}-x]/a< I_{p}(x)<[ (x^{p}+b )^{1/p}-x]/b$\end{document} for all x>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x>0$\end{document} and p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p>1$\end{document}, where Ip(x)=exp∫x∞e−tpdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{p}(x)=e^{x^{p}}\int_{x}^{\infty}e^{-t^{p}}\,dt$\end{document} is the incomplete gamma function.
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