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}ψ(x+1/3)\right)<\frac{Γ(x)}{x^xe^{-x}\sqrt{2π}}<\exp\left(-\frac{1}{2}ψ(x)\right). \end{equation*} We also prove the following new inequalities: For $x\ge 1$ \[ \sqrt{2π}x^xe^{-x}\left(x^2+\frac{x}{3}+a_*\right)^{\frac{1}{4}}<Γ(x+1)<\sqrt{2π}x^xe^{-x}\left(x^2+\frac{x}{3}+a^*\right)^{\frac{1}{4}} \] with the best possible constants $a_{\ast }=\frac{e^4}{4{\pi }^2}-\frac{4}{3}=\mathrm{0.049653963176...}$, and $a^{\ast }=1/18=\mathrm{0.055555...}$, and for $x\ge 0$ \begin{equation*} \exp\left[xψ\left(\frac{x}{\log (x+1)}\right)\right]\leqΓ(x+1)\leq\exp\left[xψ\left(\frac{x}{2}+1\right)\right], \end{equation*} where $\psi$ is the digamma function.