I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality:
consider 9 numbers $a_1,a_2,a_3,b_1,b_2,b_3,c_1,c_2,c_3$, sum of $a$'s is at most 1, and the same with $b$ and $c$. first inequality is $\sum_{i\neq j\neq k \in \{1,2,3\}} \frac{a_ib_jc_k}{3- b_i - c_i - a_j - c_j - a_k - b_k}\left( \frac{2}{2-b_k-c_j}+ \frac{2}{2-a_k-c_i} + \frac{2}{2-a_j-b_i}\right) \leq (a_1+a_2+a_3)(b_1+b_2+b_3)(c_1+c_2+c_3)$
The second I don't know how to show yet. I've written a program checking it for all possible configurations of numbers, and it seems like it also should be true. Still, I don't know how to get a hold on it because here i have terms once negative and once positive and thus i cannot use AM-HM anymore to split terms nicely. Can someone point me to similar inequalities that resemble the inclusion exclusion formula? Any help would be much appreciated. Here's the inequality: $1-\sum a_i -\sum b_j -\sum c_k + \sum_{i\neq j \in\{1,2,3\}} \frac{2a_ib_j}{2-b_i - a_j} + \sum_{j\neq k \in\{1,2,3\}} \frac{2b_jc_k}{2-c_j - b_k} + \sum_{i\neq k \in\{1,2,3\}} \frac{2a_ic_k}{2-c_i - a_k} - \sum_{i\neq j\neq k \in \{1,2,3\}} \frac{a_ib_jc_k}{3- b_i - c_i - a_j - c_j - a_k - b_k}\left( \frac{2}{2-b_k-c_j}+ \frac{2}{2-a_k-c_i} + \frac{2}{2-a_j-b_i}\right) \leq (1-a_1-a_2-a_3)(1-b_1-b_2-b_3)(1-c_1-c_2-c_3)$
