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Gurland’s ratio for the gamma function. (English) Zbl 1081.33005
The author gives several new and interesting results. More pricesely, the author considers the ratio \[ T(x, y) = {\Gamma(x)\Gamma(y) \over \Gamma^2\left(\frac{x + y}{2}\right)} \] 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 \({x \to \frac{x}{(1 - e^{-x})}}\) 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 \(\mathbb R^n\) and also to the problem of finding the inverse of the function \({x \to T\left(\frac{1}{x}, \frac{3}{x}\right)}\), which is very important in applied statistics. This is a very interesting paper.

MSC:
33B15 Gamma, beta and polygamma functions
26D07 Inequalities involving other types of functions
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