Sharp inequalities related with Burnside's formula

Abstract

We prove the following double inequality related with Burnside's formula for $n!$ \begin{equation*} \sqrt{2π}\left(\frac{n+a_*}{e}\right)^{n+a_*}<n!<\sqrt{2π}\left(\frac{n+a^*}{e}\right)^{n+a^*}\,(n\in\mathbb{N}), \end{equation*} where the constants $a_{\ast }=\mathrm{0.428844044...}$ and $a^{\ast }=0.5$ are the best possible. We believe that the method we used in the proof gives insight to undergraduate students to understand how simple inequalities can be established.

appeared iin Proyyecioes Journal of Mathematics