Sharp inequalities related with Burnside's formula

Abstract

We prove the following double inequality related with Burnside's formula for $n!$ \begin{equation*} \sqrt{2\pi}\left(\frac{n+a_*}{e}\right)^{n+a_*}<n!<\sqrt{2\pi}\left(\frac{n+a^*}{e}\right)^{n+a^*}\,(n\in\mathbb{N}), \end{equation*} where the constants $a_*=0.428844044...$ and $a^*=0.5$ 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.