Gurland's ratio for the gamma function
We consider the ratio T(x, y) = Gamma(x)Gamma(y)/Gamma(2)((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 axe applied to a problem related to the volume of the unit ball in R-n 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. (c) 2005 Elsevier Ltd. All rights reserved.
gurland's ratio gamma function polygamma functions cramer-rao inequality inverse function