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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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##### References:
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