Sharp Inequalities for Factorial N

Abstract

Let n be a positive integer. We prove nn+1e-n√2π/√n-α ≤ n! < nn+1e-n2√2π/√n-β with the best possible constants α = 1- 2π-2 = 0.149663... and β = 1/6 = 0.1666666... This refines and extends a result of Sandor and Debnath, who proved that the double inequality holds with α = 0 and β = 1.