Gurlandʹs ratio for the gamma function

We consider the ratio T(x, y) = г(x)г(y) / г2((x + y)/2) and its properties related to convexity, logarithmic convexity, Schur-convexity, and complete monotonicity. Several new bounds and asymptotic expansions for T are derived. Sharp bounds for the function x → x/(1 - e−x) are presented, as well as bounds for the trigamma function. The results are applied to a problem related to the volume of the unit ball in Rn and also to the problem of finding the inverse of the function x → T(1/x, 3/x), which is of importance in applied statistics.