GAUTSCHI'S INEQUALITY:

\(\bullet\) Let \(x\in\mathbb{R}^+\) and \(s\in(0,1)\). Then, an inequality for ratios of gamma functions known as Gautschi's inequality:
\[x^{1-s}<\dfrac{\Gamma(x+1)}{\Gamma(x+s)}<(x+1)^{1-s}\]
\(\bullet\) Asymptotic behaviour of the ratios of gamma functions:
\[\displaystyle\lim_{x\rightarrow\infty}\dfrac{\Gamma(x+1)}{\Gamma(x+s)x^{1-s}}=1\]
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