Complete monotonicity of a ratio of gamma functions and some combinatorial inequalities for multinomial coefficients
Let $m,n\in \mathbb{N}$. For all $i\in \{1,2,\dots,m\}$, $j\in \{1,2,\dots,n\}$, choose constants $X_i,Y_j > 0$, $M\in (0,1]$ and let $(x_{ij})\in (0,\infty)^{m\times n}$ be such that $\sum_{j=1}^n x_{ij} = X_i$, $\sum_{i=1}^m x_{ij} = Y_j$ and $\sum_{i=1}^m X_i = \sum_{j=1}^n Y_j = M$. We prove that the ratio of gamma functions \begin{equation*} a \mapsto \frac{\prod_{i=1}^m \Gamma(a X_i + 1) \prod_{j=1}^n \Gamma(a Y_j + 1)}{\prod_{i=1}^m \prod_{j=1}^n \Gamma(a x_{ij} + 1)} \end{equation*} is logarithmically completely monotonic on $(0,\infty)$. This result complements the logarithmically complete monotonicity of multinomial probabilities shown in Ouimet (2018); Qi et al. (2018), and the recent survey of Qi & Agarwal (2019) on the complete monotonicity of functions related to ratios of gamma functions. As a consequence of the log-convexity, we obtain new combinatorial inequalities for multinomial coefficients.