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

TL;DR

This paper establishes new two-sided bounds for Stirling-type asymptotics of the pi and gamma functions, improving previous results through elementary proofs and numerical validation.

Contribution

It introduces integral identity versions of Gautschi's inequality and derives asymptotically optimal bounds for the pi and gamma functions using elementary methods.

Findings

Bounds are asymptotically optimal.

Numerical comparisons demonstrate effectiveness.

Connects and generalizes previous results.

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.