Let $\mathcal{E}_n$ be the set of all $n\times n$ permutation matrices that exchange two components. Such matrices have the form $$ \begin{bmatrix} \ddots& \\ &0 && 1 \\ &&\ddots \\ &1 && 0 \\ &&&&\ddots \end{bmatrix} $$ A matrix $T$ is called a Pigou-Dalton 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 by $y$
2. $x=Dy$ for a doubly stochastic matrix $D$
3. $x=T_1T_2\cdots T_k y$ for finitely many PDT $T_1,\ldots, T_k$
4. $\sum_{i=1}^n \theta(x_i) \leq \sum_{i=1}^n \theta(y_i)$ for all convex function $\theta$
5. $x$ lies in the convex hull whose vertex set is $$ \big\{ (y_{\pi(1)},y_{\pi(2)},\ldots, y_{\pi(n)}):\, \pi \text{ is a permutation of }\{1,\ldots, n\}\big\}. $$
6. For any $n$ non-negative real numbers $a_1,\ldots, a_n$ $$ \sum_{\pi} a_1^{x_{\pi(1)}} a_2^{x_{\pi(2)}} \cdots a_n^{x_{\pi(n)}} \leq \sum_{\pi} a_1^{y_{\pi(1)}} a_2^{y_{\pi(2)}} \cdots a_n^{y_{\pi(n)}} $$ where summation is taken over all permutations of $\{1,\ldots, n\}$

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.