I need to prove this inequality, but I do not have a good background in algebra, if you can guide me:
We have: $$ p_1+p_2+\ldots+p_k = q_1+q_2+\ldots+q_k+\ldots+q_t = n $$ and $$ p_1 + 2p_2 + \ldots +kp_k < q_1 + 2q_2 + \ldots +kq_k+(k+1)q_{k+1}+\ldots+tq_t $$ and $$ 1 \leq l_1 < l_2 < \ldots < l_k < \ldots < l_t $$ and $$ p_1 \geq p_2 \geq ... \geq p_k\\ $$ and $$ q_1 \geq q_2 \geq ... \geq q_t $$ where
- $p_i \in \mathbb{N^*}$,
- $q_i \in \mathbb{N^*}$,
- $l_i\in \mathbb{N^*}$ for all $i$.
Then I need to prove that: $$ l_1p_1+l_2p_2+\ldots +l_kp_k < l_1q_1+l_2q_2+\ldots+l_kq_k+\ldots+l_tq_t $$
If anyone can help me please.
Best regards
After several mouths of research, i found a counter example which i will share with you:
For $k = 5$: $p_1 = 6, p_2 = 4, p_3 = 4, p_4 = 1, p_5 = 1$
For $t = 6$: $q_1 = 6, q_2 = 6, q_3 = 1, q_4 = 1, q_5 = 1, q_6 = 1$
We take $l_1 = 1, l_2 = 2, l_3 = 5, l_4 = 6, l_5 = 7, l_6 = 8$
So we get: $\sum_{i=1}^{5} ip_i = 35$ and $\sum_{i=1}^{6} iq_i = 36$
and : $\sum_{i=1}^{5} l_ip_i = 47 > \sum_{i=1}^{6} l_iq_i = 44$