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For $m,n\in \mathbb{N}$, let $0 < \alpha_i,\beta_j,\lambda_{ij} \leq 1$ besuch that $\sum_{j=1}^n \lambda_{ij} = \alpha_i$, $\sum_{i=1}^m \lambda_{ij} =\beta_j$, and $\sum_{i=1}^m \alpha_i = \sum_{j=1}^n \beta_j \leq 1$. We provethat the ratio of gamma functions \begin{equation*} \hspace{-15mm}t \mapsto\frac{\prod_{i=1}^m \Gamma(\alpha_i t + 1) \prod_{j=1}^n \Gamma(\beta_j t +1)}{\prod_{i=1}^m \prod_{j=1}^n \Gamma(\lambda_{ij} t + 1)} \end{equation*} islogarithmically completely monotonic on $(0,\infty)$. This result complementsthe logarithmically complete monotonicity of multinomial probabilities shown inOuimet (2018), Qi et al (2018), and the recent survey of Qi & Argawal (2019) onthe complete monotonicity of functions related to ratios of gamma functions. Asa consequence of the log-convexity, we obtain new combinatorial inequalitiesfor multinomial coefficients.

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