Abstract
We consider the ratio T( x, y) = г( x)г( y) / г 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 are 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.
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