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Gamma function and inequality [entire topic moved to new forum]

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Let \( \Gamma \) be  Euler's Gamma function. Prove that:

$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > 0,\; 0 < s < 1$$

 

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