Sum of powers inequality?

Question

We have numbers $a_i \ge 0$ and $b_i\ge 0$ with $i=1,2,\ldots,n$, and we know $$ \sum_{i=1}^n a_i = \sum_{i=1}^n b_i>0. $$ Now if $$ \sum_{i=1}^n a_i^2 \ge \sum_{i=1}^n b_i^2, $$ can one show formally that $$ \sum_{i=1}^n a_i^k \ge \sum_{i=1}^n b_i^k $$ for $k>2$?

Answer 1

Not true.
Consider $a = (0,0,0.5,0.5)$, $b = (0.106, 0.106, 0.106, 0.682)$.
Then $$ \eqalign{\sum_i a_i &= \sum_i b_i = 1 \cr \sum_i a_i^2 = 0.5 &\ge \sum_i b_i^2 = 0.498832\cr \sum_i a_i^3 = 0.25 &< \sum_i b_i^3 = 0.320787616\cr}$$