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

In this paper, we prove that for a≥31/98\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\ge 31/98$$\end{document}, the function fax=lnΓx+12-xlnx+x-12ln2π+x24x2+a-7/120x4+ax2+98a-31/1680\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_{a}\left( x\right) =\ln \Gamma \left( x+\frac{1}{2}\right) -x\ln x+x-\frac{ 1}{2}\ln \left( 2\pi \right) +\dfrac{x}{24}\dfrac{x^{2}+a-7/120}{ x^{4}+ax^{2}+\left( 98a-31\right) /1680} \end{aligned}$$\end{document}is strictly increasing (decreasing) and concave (convex) on 0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( 0,\infty \right) $$\end{document} if and only if a≥5281/6068\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\ge 5281/6068$$\end{document} (a=31/98\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=31/98$$\end{document}). Moreover, we show that the necessary and sufficient condition for function Fax=-x4+ax2+98a-311680fax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_{a}\left( x\right) =-\left( x^{4}+ax^{2}+\frac{98a-31}{1680}\right) f_{a}\left( x\right) \end{aligned}$$\end{document}for a∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in \mathbb {R}$$\end{document} to be completely monotonic on 0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( 0,\infty \right) $$\end{document} is also a≥5281/6068\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\ge 5281/6068$$\end{document}. These yield some new sharp bounds for the gamma function.