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{equation*} \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{equation*} We also prove the following new inequalities: For $x\geq1$ \[ \sqrt{2\pi}x^xe^{-x}\left(x^2+\frac{x}{3}+a_*\right)^{\frac{1}{4}}<\Gamma(x+1)<\sqrt{2\pi}x^xe^{-x}\left(x^2+\frac{x}{3}+a^*\right)^{\frac{1}{4}} \] with the best possible constants $a_*=\frac{e^4}{4\pi^2}-\frac{4}{3}=0.049653963176...$, and $a^*=1/18=0.055555...$, and for $x\geq0$ \begin{equation*} \exp\left[x\psi\left(\frac{x}{\log (x+1)}\right)\right]\leq\Gamma(x+1)\leq\exp\left[x\psi\left(\frac{x}{2}+1\right)\right], \end{equation*} where $\psi$ is the digamma function.