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Monotonicity results for the Gamma function. (English) Zbl 1059.33004
After exhaustive background description, the authors offer the following results. The fuctions $$x\mapsto \Gamma(x+1)^{1/x}(x+1)^{-1/2}$$ and $$x\mapsto \Gamma(x+1)^{1/x}x^{-1/2}$$ are strictly increasing on $$[1,\infty[$$ or $$[2,\infty[,$$ respectively, while $$x\mapsto \Gamma(x+1)^{1/x}(x+1)^{-1}$$ is strictly decreasing on $$[1,\infty[.$$ The paper closes with 29 references.

##### MSC:
 33B15 Gamma, beta and polygamma functions 26A48 Monotonic functions, generalizations 26D07 Inequalities involving other types of functions
##### Keywords:
gamma function; monotonicity; inequalities
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