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

For $m,n\in \mathbb{N}$, let $0 < \alpha_i,\beta_j,\lambda_{ij} \leq 1$ be such that $\sum_{j=1}^n \lambda_{ij} = \alpha_i$, $\sum_{i=1}^m \lambda_{ij} = \beta_j$, and $\sum_{i=1}^m \alpha_i = \sum_{j=1}^n \beta_j \leq 1$. We prove that 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*} 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 & Argawal (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.