Consider $\{p_i \in [0,c]\}_i$, $c<1$, such that $\sum_{i=1}^{k} p_i = P$ for some positive $k>2$ and constant $P < 1$. Let $(x_i \in \mathbb{N})_i$. Is $$f(p_1,\ldots,p_k) = \sum_{x_1,x_2,\ldots,x_k=0}^{c} \frac{p_1^{x_1}}{x_1} \frac{p_2^{x_2}}{x_2!} \cdots \frac{p_k^{x_k}}{x_k!} $$ Schur-concave in $\{p_i\}$? The function is symmetric, but it is a bit hard to check the cond $\partial f/\partial p_i - \partial f/\partial p_j$. Any suggestion?
PS: The function is similar to the marginal of a Multinomial distribution.