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Homemajorization
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majorization
For any real vector $x=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}$, let $x_{{(1)}}\geq x_{{(2)}}\geq\cdots\geq x_{{(n)}}$ denote the components of $x$ in nonincreasing order.
For $x,y\in\mathbb{R}^{n}$, we say that $x$ is majorized by $y$, or $y$ majorizes $x$, if
$\displaystyle\sum_{{i=1}}^{m}x_{{(i)}}$  $\displaystyle\leq\sum_{{i=1}}^{m}y_{{(i)}},\quad\text{ for $m=1,\ldots,n1$, and}$  
$\displaystyle\sum_{{i=1}}^{n}x_{{(i)}}$  $\displaystyle=\sum_{{i=1}}^{n}y_{{(i)}}$ 
A common notation for “$x$ is majorized by $y$” is $x\prec y$.
Remark:
A canonical example is that, if $y_{1}$, $y_{2},\ldots,y_{n}$ are nonnegative real numbers such that their sum is equal to 1, then
$\left(\frac{1}{n},\ldots,\frac{1}{n}\right)\prec(y_{1},\ldots,y_{n}).$ 
In general, $x\prec y$ vaguely means that the components of $x$ is less spread out than are the components of $y$.
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.
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