characterizations of majorization
Let ${\mathcal{E}}_{n}$ be the set of all $n\times n$ permutation matrices^{} that exchange two components^{}. Such matrices have the form
$$\left[\begin{array}{cc}\hfill \mathrm{\ddots}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill 0\hfill & \hfill \hfill & \hfill 1\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \mathrm{\ddots}\hfill \\ \hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \mathrm{\ddots}\hfill \end{array}\right]$$ 
A matrix $T$ is called a PigouDalton transfer (PDT) if
$$T=\alpha I+(1\alpha )E$$ 
for some $\alpha $ between 0 and 1, and $E\in {\mathcal{E}}_{n}$.
The following are equivalent^{}

1.
$x$ is majorized (http://planetmath.org/Majorization^{}) by $y$.

2.
$x=Dy$ for a doubly stochastic matrix $D$.

3.
$x={T}_{1}{T}_{2}\mathrm{\cdots}{T}_{k}y$ for finitely many PDT ${T}_{1},\mathrm{\dots},{T}_{k}$.

4.
${\sum}_{i=1}^{n}\theta ({x}_{i})\le {\sum}_{i=1}^{n}\theta ({y}_{i})$ for all convex function $\theta $.

5.
$x$ lies in the convex hull whose vertex set is
$$\{({y}_{\pi (1)},{y}_{\pi (2)},\mathrm{\dots},{y}_{\pi (n)}):\pi \text{is a permutation of}\{1,\mathrm{\dots},n\}\}.$$ 
6.
For any $n$ nonnegative real numbers ${a}_{1},\mathrm{\dots},{a}_{n}$,
$$\sum _{\pi}{a}_{1}^{{x}_{\pi (1)}}{a}_{2}^{{x}_{\pi (2)}}\mathrm{\cdots}{a}_{n}^{{x}_{\pi (n)}}\le \sum _{\pi}{a}_{1}^{{y}_{\pi (1)}}{a}_{2}^{{y}_{\pi (2)}}\mathrm{\cdots}{a}_{n}^{{y}_{\pi (n)}}$$ where summation is taken over all permutations^{} of $\{1,\mathrm{\dots},n\}$.
The equivalence of the above conditions are due to Hardy, Littlewood, Pólya, Birkhoff, von Neumann and Muirhead.
Reference

•
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities^{}, 2nd edition, 1952, Cambridge University Press, London.

•
A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, 1979, Acadamic Press, New York.
Title  characterizations of majorization 

Canonical name  CharacterizationsOfMajorization 
Date of creation  20130322 15:26:37 
Last modified on  20130322 15:26:37 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  7 
Author  Mathprof (13753) 
Entry type  Theorem 
Classification  msc 26D99 
Related topic  BirkoffVonNeumannTheorem 
Related topic  MuirheadsTheorem 
Defines  PigouDalton transfer 