There are two sets $\{\lambda_1,\lambda_2,...,\lambda_N\}$ with $\lambda_1\geq\lambda_2\geq...\geq\lambda_N,\sum_{i=1}^N\lambda_i=1$, $\{a_1,a_2,...,a_N\}$ with $a_1 \leq a_2\leq...\leq a_N$ and a permutation $P$ on $\{1,2,...,N\}$.
I want to know if $\sum_{i=1}^N \lambda_i a_i \leq \sum_{i=1}^N \lambda_i a_{P^{-1}(i)}$ holds for any permutation $P$?
It is obvious that if the permutation is just exchangeing the elements, i.e. $P(i)=j$ and $P(j)=i$, then the inequality holds. For more general case, I tried to compare them by $\sum_{i=1}^N \lambda_i a_i - \sum_{i=1}^N \lambda_i a_{P^{-1}(i)}$ and dividing the index set into three parts: $I_1=\{i:i>P^{-1}(i)\},I_2=\{i:i<P^{-1}(i)\},I_3=\{i:i=P^{-1}(i)\}$, but I don't know how to proceed.