Bounds for the Gamma Function

Abstract

We improve the upper bound of the following inequalities for the gamma function $\Gamma$ due to H. Alzer and the author.

$$\begin{aligned} \exp \left( -\frac{1}{2}\psi (x+1/3)\right)<\frac{\Gamma (x)}{x^xe^{-x}{\sqrt{2\pi }}} <\exp \left( -\frac{1}{2}\psi (x)\right) . \end{aligned}$$

We also prove the following new inequalities: for $x\ge 1$

$$\begin{aligned} {\sqrt{2\pi }}x^xe^{-x}\left( x^2+\frac{x}{3}+a_*\right) ^{\frac{1}{4}}<\Gamma (x+1)<{\sqrt{2\pi }}x^xe^{-x}\left( x^2+\frac{x}{3}+a^*\right) ^{\frac{1}{4}} \end{aligned}$$

with the best possible constants $a_*=\frac{e^4}{4\pi ^2}-\frac{4}{3}=0.049653963176\ldots$, and $a^*=1/18=0.055555\ldots$, and for $x\ge 0$

$$\begin{aligned} \exp \left[ x\psi \left( \frac{x}{\log (x+1)}\right) \right] \le \Gamma (x+1)\le \exp \left[ x\psi \left( \frac{x}{2}+1\right) \right] , \end{aligned}$$

where $\psi$ is the digamma function.