Some inequalities for Gurland's ratio of the gamma functions

TL;DR

This paper introduces a modified Gurland ratio for gamma functions, derives new inequalities and bounds using analytic expansions, and explores the asymptotic behavior of gamma function ratios.

Contribution

It presents a modified form of the Gurland ratio, derives finite expansions and bounds, and establishes new inequalities extending classical results.

Findings

Derived a finite expansion involving the Hurwitz zeta function.

Established explicit upper bounds for the remainder term.

Obtained new bilateral inequalities for the Gurland ratio.

Abstract

This paper investigates the classical Gurland ratio of the gamma function and introduces its modified form, $\mathcal{G}^{\star}(x,y)$, which is particularly amenable to analytic expansions. By utilizing the Weierstrass product representation of the gamma function, we derive a finite expansion for the logarithm of $\mathcal{G}^{\star}(x,y)$ involving the Hurwitz zeta function. Explicit upper bounds for the remainder term are established, providing a rigorous basis for convergence analysis. As a direct consequence, we obtain new bilateral inequalities for the Gurland ratio and demonstrate the existence of a specific parameter $t(x,y)$ related to the Mean Value Theorem. Furthermore, we formulate open problems regarding the optimal localization of this parameter. These results extend the classical works of Gurland, Gautschi, and Merkle, offering new insights into the asymptotic behavior of gamma function ratios.