A Two-Sided Inequality of Gamma Function

Abstract:

      This note shows that the inequality \[ (\frac{x}{e})^x\sqrt {2\pi x} (1+\frac{1}{12x})<\Gamma (x+1)<(\frac{x}{e})^x\sqrt {2\pi x} (1+\frac{1}{12x-0.5})\] holds for all $x\ge 1$.