Complete monotonicity of a ratio of gamma functions and some combinatorial inequalities for multinomial coefficients

Frédéric Ouimet

For $m,n\in \mathbb{N}$, let $0<{\alpha }_i,{\beta }_j,{\lambda }_{ij}\le 1$ be such that ${\sum }_{j=1}^n{\lambda }_{ij}={\alpha }_i$, ${\sum }_{i=1}^m{\lambda }_{ij}={\beta }_j$, and ${\sum }_{i=1}^m{\alpha }_i={\sum }_{j=1}^n{\beta }_j\le 1$. We…