Bounds for the gamma function

TL;DR

This paper refines existing bounds for the gamma function and introduces new inequalities with optimal constants, enhancing the accuracy of gamma function approximations for various ranges of x.

Contribution

The paper improves upper bounds for the gamma function and establishes new inequalities with optimal constants, extending the theoretical understanding of gamma function bounds.

Findings

Improved upper bounds for gamma function involving digamma functions.

New inequalities with best possible constants for gamma function estimates.

Enhanced bounds applicable for different ranges of x.

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.