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

Abstract

In this paper, we prove that for $a\ge 31/98$, the function

$$\begin{aligned} f_{a}\left( x\right) =\ln \Gamma \left( x+\frac{1}{2}\right) -x\ln x+x-\frac{ 1}{2}\ln \left( 2\pi \right) +\dfrac{x}{24}\dfrac{x^{2}+a-7/120}{ x^{4}+ax^{2}+\left( 98a-31\right) /1680} \end{aligned}$$

is strictly increasing (decreasing) and concave (convex) on $\left( 0,\infty \right)$ if and only if $a\ge 5281/6068$ ($a=31/98$). Moreover, we show that the necessary and sufficient condition for function

$$\begin{aligned} F_{a}\left( x\right) =-\left( x^{4}+ax^{2}+\frac{98a-31}{1680}\right) f_{a}\left( x\right) \end{aligned}$$

for $a\in \mathbb {R}$ to be completely monotonic on $\left( 0,\infty \right)$ is also $a\ge 5281/6068$. These yield some new sharp bounds for the gamma function.