🚀 go-pugleaf

Can the following theorem be easily proofed?

Theorem:
Let 
    1. p_1\geq p_2\geq\ldots\geq p_n and q_1\geq q_2\geq\ldots\geq q_n
       with 0\leq p_i\leq 1 and 0\leq q_i\leq 1 \forall 1\leq i\leq n
    2. \sum_{i=1}^np_i=\sum_{i=1}^nq_i=1
       (imagine p_i and q_i as probabilities in decreasing order)       
    3. \sum_{i=1}^kp_i\geq\sum_{i=1}^kq_i \forall 1\leq k\leq n
       (the cumulated sum of p_i is always greater or equal than the 
        cumulated sum of q_i)
Then
    \sum_{i=1}^np_i(1-q_i)^\alpha\leq\sum_{i=1}^nq_i(1-p_i)^\alpha
    \forall real \alpha\geq 1 

If this is true, it follows that a weaker expert can better predict a
stronger expert than vice versa.
Thanks in advance. 		  

Bernhard Strohmeier
bstrohm@erasmus.hrz.uni-bielefeld.de