Convex combination of partial sums of integers

Question

I have a vector $(v_1,...,v_k)$ of non-negative integers, each entry being unkown, and an unkown function $f:\mathbb{Z}\to (0,1)$ such that $f(1)+...+f(k) = 1$. The only known quantity is $\sum_{i=1}^k v_i=E$. I try to find an upper bound $E'<E$ for the sum $$\sum_{i=1}^k f(i)\sum_{j\neq i} v_j.$$ The numbe $E$ itself is an upper bound since $\sum_{j\neq i} v_j\leq \sum_{j=1}^k v_j$. Since I am basically considering convex combinations of the partial sums $\sum_{j\neq i} v_j$ I hoped to do better. Unfortunately, I get stuck at a term $$\sum_{i=1}^k f(i)\sum_{j\neq i} v_j = E - \sum_{i=1}^k f(i)v_i$$ since all estimates I know give upper bounds for $\sum_{i=1}^k f(i)v_i$ like Jensen inequality and a I am struggeling with a lower bound since $f$ might be really small and some $v_i$ even $0$. Does anyone have an idea how I might proceed?

Answer 1

It's certainly possible for that sum to equal $E$, in which case you can't do much better. In any case, the most that can be said is that $$\sum_{i=1}^k f(i)\sum_{j\neq i} v_j\le\max_i\sum_{j\ne i}v_j.$$