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Inequalities for ratios of Gamma functions
We prove that the function $$ \log \frac{Γ(x+a+b)Γ(x)}{Γ(x+a)Γ(x+b)}-b\log\left(1+\frac{a}{x}\right) $$ is a generalized Stieltjes function of order $3$ for $a>0$ and $00$ and $b>1$.
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