I am hoping to prove the following:
Let $\{a_i\}_{i=1}^n$, $\{b_i\}_{i=1}^n$ be sequences with non-negative terms. Suppose that they satisfy $\sum_{i=1}^na_i = \sum_{i=1}^nb_i$ and $\sum_{i=1}^ja_i \geq \sum_{i=1}^jb_i$ for any $1 \leq j \leq n$. Then if $\{c_i\}_{i=1}^n$ is a non-decreasing sequence with non-negative terms we have $$\sum_{i=1}^n a_ic_i \leq \sum_{i=1}^n b_ic_i$$
I will use proof by induction.
Base case: In this case $a_1c_1 \leq b_1c_1$ trivially holds.
Inductive hypothesis: Assume the sequences satisfy the requirements and result for some $n = m$. We show the statement for $m+1$ holds.
We construct sequences of length $m+1$ by choosing $a_{m+1} = a_m$, $b_{m+1} = b_m$ and letting $c_{m+1}$ be any number that satisfies $c_{m+1} \geq c_m \geq 0$.
Therefore we have $\sum_{i=1}^{m+1}a_i = \sum_{i=1}^{m+1}b_i$ and $\sum_{i=1}^ja_i \geq \sum_{i=1}^jb_i$ for any $j = 1,\ldots,m+1$.
Since $\sum_{i=1}^ma_i \geq \sum_{i=1}^mb_i$, we know $a_{m+1} \leq b_{m+1}$. We multiple through by $z_{m+1}$ to obtain $z_{m+1}a_{m+1} \leq z_{m+1}b_{m+1}$, where the inequality is preserved by non-negativity of $z_{m+1}$.
Therefore adding this inequality to the induction hypothesis we have $\sum_{i=1}^{m+1} a_ic_i \leq \sum_{i=1}^{m+1} b_ic_i$ and we are done.
I think something is wrong here because I don't explicitly use the fact that $c_i$ are non-decreasing. There are easy counter-examples when $c_i$ are decreasing.
I am also not sure if I am "allowed" to construct the sequence of length $m+1$ or if I am supposed to assume that there exists some sequence of length $m+1$ and the partial sequence up to $m$ satisfies the inductive hypothesis.