On Gautschi & Stirling Identities, Asymptotics and Inequalities for the Pi (or Gamma) Function

Abstract

We derive two-sided bounds for a class of Stirling-type asymptotic formulas for piecewise logarithmic interpolations of the pi function, and hence also for the factorials and the gamma functions. The bounds are derived by first proving some integral identity versions of Gautschi's inequality and a class of Stirling-type asymptotic formulas, and then bounding these integrals by asymptotically optimal bounds. Additionally, all the proofs given rely only on common elementary arguments and connect, generalise and possibly improve various results that have been published previously. Lastly, we provide numerical comparisons concerning the effectiveness and behaviour of the bounds and approximations in a graphical manner, which clearly indicate that the bounds are asymptotically optimal.