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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.

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  • $\begingroup$ What is the domain of $f$? Positive or also negative input? $\endgroup$ – LinAlg Nov 20 '18 at 14:45
  • $\begingroup$ edited the question accordingly. $\{p_i\}'s$ are probability simplex. $\endgroup$ – Jeff Nov 20 '18 at 14:57
  • $\begingroup$ then the answer is trivial, right? $\endgroup$ – LinAlg Nov 20 '18 at 15:15
  • $\begingroup$ you were right. Actually, it should be Schur-concave, and also I had a summation, which was missing in the original question. Now, it is not trivial to me whether it is Schur-concave or not... $\endgroup$ – Jeff Nov 20 '18 at 18:56

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