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.