  Let $d\in \mathbb{N}$ and let $\gamma_i\in [0,\infty)$, $x_i\in (0,1)$ be
such that $\sum_{i=1}^{d+1} \gamma_i = M\in (0,\infty)$ and $\sum_{i=1}^{d+1}
x_i = 1$. We prove that \begin{equation*} a \mapsto \frac{\Gamma(aM +
1)}{\prod_{i=1}^{d+1} \Gamma(a \gamma_i + 1)} \prod_{i=1}^{d+1} x_i^{a\gamma_i}
\end{equation*} is completely monotonic on $(0,\infty)$. This result
generalizes the one found by Alzer (2018) for binomial probabilities ($d=1$).
As a consequence of the log-convexity, we obtain some combinatorial
inequalities for multinomial coefficients. We also show how the main result can
be used to derive asymptotic formulas for quantities of interest in the context
of statistical density estimation based on Bernstein polynomials on the
$d$-dimensional simplex.
