Majorization and Schur-convexity

Majorization 

A real vector $b = (b_1,\dotsc,b_n)$ is said to majorize $a = ( a_1,\dotsc, a_n)$, denote $a \succ b\ $  if

1.  $\sum_{i=1}^n a_i  = \sum_{i=1}^n b_i$, and
2. $\sum_{i=k}^n a_{(i)} \leq \sum_{i=k}^n b_{(i)}$, $k = 2,\dotsc,n$

where $a_{(1)} \leq \dotsb \leq a_{(n)}$, $b_{(1)} \leq \dotsb \leq b_{(n)}$ are $a$ and $b$ arranged in increasing order. 

A function $\phi(a)$ symmetric in the coordinates of $a = ( a_1, \dotsc, a_n )$ is said to be Schur-concave if  $a \succ b$ implies $\phi(a) \ge \phi(b)$.  

A function $\phi(a)$ is Schur-convex if $-\phi(a)$ is Schur-concave.