We have three sequences of positive integers $l$, $p$ and $q$ such that: $$ p_1 \geq p_2 \geq \cdots \geq p_k\ \ and\ q_1 \geq q_2 \geq \cdots \geq q_k \geq \cdots \geq q_h \ \ where:\ \ k < h $$
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
$$ p_1+p_2+\cdots+p_k = q_1+q_2+\cdots+q_k+\cdots+q_h=n $$
Is the following statement true:
$$ l_1(p_1-q_1)+l_2(p_2-q_2)+\cdots+l_k(p_k-q_k) \leq l_k(p_1-q_1)+l_k(p_2-q_2)+\cdots+l_k(p_k-q_k) $$
This proof will help me to prove another result that uses it in its proof.
Best regards
A counterexample is given by $k=2$, $h=3$, $n=18$, $$p_1=9, p_2=9, q_1=10, q_2=4, q_3=4, l_1=1, l_2=2, l_3=2.$$