Inequalities for ratios of Gamma functions

Abstract

We prove that the function $$\log \frac{\Gamma (x+a+b)\Gamma (x)}{\Gamma (x+a)\Gamma (x+b)}-b\log \left(1+\frac{a}{x}\right)$$ is a generalized Stieltjes function of order $3$ for $a>0$ and $0<b<1$. A similar result is obtained when $a>0$ and $b>1$.