# Schur-convexity of multinomial distribution

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.

• What is the domain of $f$? Positive or also negative input? – LinAlg Nov 20 '18 at 14:45
• edited the question accordingly. $\{p_i\}'s$ are probability simplex. – Jeff Nov 20 '18 at 14:57
• then the answer is trivial, right? – LinAlg Nov 20 '18 at 15:15
• 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... – Jeff Nov 20 '18 at 18:56