Sharp inequalities related with Burnside's formula

TL;DR

This paper establishes sharp bounds for factorials using refined constants in Burnside's formula, providing insights into inequality proofs suitable for undergraduate learning.

Contribution

It introduces the best possible constants for double inequalities related to Burnside's formula for factorials, enhancing understanding of factorial approximations.

Findings

Established sharp bounds for n! with optimal constants

Provided a method accessible for undergraduate students to understand inequality proofs

Enhanced the theoretical understanding of factorial approximations

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.