Sharp Smith’s bounds for the gamma function

Abstract

Abstract Among various approximation formulas for the gamma function, Smith showed that Γ(x+12)∼S(x)=2π(xe)x(2xtanh12x)x/2,x→∞, $$ \Gamma \biggl( x+\frac{1}{2} \biggr) \thicksim S ( x ) =\sqrt{2 \pi } \biggl( \frac{x}{e} \biggr) ^{x} \biggl( 2x\tanh \frac{1}{2x} \biggr) ^{x/2}, \quad x\rightarrow \infty , $$ which is a little-known but accurate and simple one. In this note, we prove that the function x↦lnΓ(x+1/2)−lnS(x) $x\mapsto \ln \Gamma ( x+1/2 ) - \ln S ( x ) $ is strictly increasing and concave on (0,∞) $( 0,\infty ) $, which shows that Smith’s approximation is just an upper one.

Keywords

• Gamma function
• Approximation formula
• Inequality