We have three sequences of positive integers $l$, $p$ and $q$ such that: $$ p_1 \geq p_2 \geq \cdots \geq p_k\text{ and } q_1 \geq q_2 \geq \cdots \geq q_k \geq \cdots \geq q_h \text{ where: } k < h $$
and
$$ 0\leq l_1 \leq l_2 \cdots \leq l_k\leq \cdots \leq l_h $$
and $$ p_1+2p_2+\cdots+kp_k < q_1+2q_2+\cdots+kq_k+\cdots+hq_h $$
and
$$ p_1+p_2+\cdots+p_k = q_1+q_2+\cdots+q_k+\cdots+q_h=n $$
I search an upper bound for this formula:
$$ l_1(p_1-q_1)+l_2(p_2-q_2)+\cdots+l_k(p_k-q_k) $$
If we put $r_i = p_i-q_i$, we can re-write it as:
$$ l_1r_1+l_2r_2+\cdots+l_kr_k $$
We distinguish 3 cases:
$$ r_i=r^{+}_i\text{ if } r_i>0 $$
$$ r_i=r^{-}_i\text{ if } r_i<0 $$
$$ r_i=r^0_i\text{ if } r_i=0 $$
So, we can find a first upper bound like that:
$$ l_1r_1+l_2r_2+\cdots+l_kr_k \leq \sum_{i\in I^{+}} l_ir^{+}_i $$
such that $I^{+}$ is the set of indices where $r_i>0$.
For my work, i need a better upper bound using the presented parameters.
Additional informations may be useful:
$$ \sum_{i=1}^h r_i = 0 $$
$$ \sum_{i=1}^k r_i = -\sum_{i=k+1}^h r_i $$
The upper bound is needed to prove another result that uses it in its proof.
Best regards
