I have $\lambda, \mu$ as 2 partitions of $n \ (=|\lambda|=|\mu|)$, and I have been given that $M \lambda=\mu$, where M is a $n \times n$ double stochastic matrix i.e. $$\sum_im_{ij}=\sum_jm_{ij}=1$$
I have to show that $$\lambda \geq \mu \ \ \ \ \ \ \ \ (\text{i.e.} \sum_{k=1}^i\lambda_k \geq \sum_{k=1}^i\mu_k \ , \ \forall i \geq 1)$$
Now, we know that $$\sum_jm_{ij}\lambda_j=\mu_i \ \ \ \ \ \ \ \ \ i\geq1$$ For $i=1$ $$\mu_1 = \sum_jm_{1j}\lambda_j \leq \lambda_1 \Big(\sum_jm_{1j} \Big)=\lambda_1$$
But I'm unable to proceed with proving the general inequality, any help will be appreciated. This is an example from Macdonald's Symmetric Functions and Hall Polynomials
$$ \sum_{k=1}^i \mu_k = \sum_{k=1}^i (M\lambda)_k = \sum_{k=1}^i \sum_j m_{kj}\lambda_j = \sum_j \lambda_j \cdot \sum_{k=1}^i m_{kj} $$ Define $p_j = \frac 1i\sum_{k=1}^i m_{kj}$. From the hypotesis you get $p_j\le 1/i$ and $\sum_j p_j= 1$. $$ \sum_{k=1}^i \frac 1i\mu_k = \sum_j \lambda_j p_j = \sum_{k=1}^i \frac 1i\lambda_k - \sum_{k=1}^i \lambda_k \left(\frac 1i -p_k\right) +\sum_{k=i+1}^n \lambda_k p_k $$ $$ \le \sum_{k=1}^i \frac 1i\lambda_k - \lambda_i\sum_{k=1}^i \left(\frac 1i -p_k\right) +\lambda_{i+1}\sum_{k=i+1}^n p_k $$ but $\sum_{k=1}^i \left(\frac 1i -p_k\right) = 1 - \sum_{k=1}^i p_k = \sum_{k=i+1}^n p_k $, so $$= \sum_{k=1}^i \frac 1i\lambda_k -(\lambda_{i}-\lambda_{i+1})\sum_{k=i+1}^n p_k \le \sum_{k=1}^i \frac 1i\lambda_k $$