PlanetMath: Muirhead's theorem

(more info)

Math for the people, by the people.

Main Menu

Muirhead's theorem

(Theorem)

Let $0\leq s_1\leq\cdots\leq s_n$ and $0\leq t_1\leq\dots\leq t_n$ be real numbers such that

$\displaystyle \sum_{i=1}^n s_i = \sum_{i=1}^n t_i\;$ and $\displaystyle \;\; \sum_{i=1}^k s_i \leq \sum_{i=1}^k t_i\;\; (k=1,\dots,n-1)$

Then for any nonnegative numbers $x_1,\dots,x_n$ ,

$\displaystyle \sum_\sigma x_1^{s_{\sigma(1)}}\dots x_n^{s_{\sigma(n)}} \geq \sum_\sigma x_1^{t_{\sigma(1)}}\dots x_n^{t_{\sigma(n)}}$

where the sums run over all permutations $\sigma$ of $\{1,2,\dots,n\}$ .

"Muirhead's theorem" is owned by Koro.

(view preamble | get metadata)

Other names:

Muirhead's inequality

Log in to rate this entry.
(view current ratings)

Cross-references: permutations, sums, numbers, real numbers
There is 1 reference to this entry.

This is version 4 of Muirhead's theorem, born on 2002-12-11, modified 2005-02-25.
Object id is 3736, canonical name is MuirheadsTheorem.
Accessed 9215 times total.

Classification:

26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact