The main goal of this post is to prove the fore mentioned inequalities, via separation theorem of functional analysis. The main result would be a proof of the result of Birkhorff:

If , where the arrays , , then there exists non-negative numbers () with , such that

.

**1. Definition and tools
**

Let , where the arrays be arrays in . We say that majorizes , in symbol , if

- and
- for all

Examples:

- (both arrays have n terms)

To prove the theorems in this note, we will need some theorems listed here:

**Strong Separation Theorem**

Let A,B be disjoint, nonempty convex subsets of a topological vector space X. If A is compact, B is closed and X is locally convex, then there is such that

.

**Rearrangement Inequality**

Let , be two sequences of real numbers. Then

**Jensen’s inequality**

Let be a convex function. Then for real numbers such that and , we have

**Weighted AM-GM inequality**

for real numbers such that and , we have

**2. Main Theorem**

**Theorem** (Birkhoff ^{1}):

If , where the arrays , , then there exists non-negative numbers () with , such that

.

Geometrically, this means that the sequences majorized by are exactly the convex hull formed by points with coordinates being all the permutations of .

**Proof of Birkhoff’s theorem** ^{2}

Let , where the arrays be arrays in . Assume that , but does not lie in the convex hull of the points for all . By separation theorem (take A to be and B to be ), we can find a linear functional such that

Note that a linear functional of is of the form for some constants . By adding a constant if needed, we see that

for some constants . The constant can be removed from this inequality.

Let be an rearrangement of . By rearrangement inequailty, we have

.

The last step is because

by rearrangement inequality.

This is a contradiction, since

**3. Application 1: Karamata inequality (Majorization inequality)**

**Theorem **(Karamata)

If , where the arrays , , then for any convex function , we have

**Proof**

By Birkhoff’s theorem, we have

.

By Jensen’s inequality,

**3. Application 2: Muirhead inequality**

**Theorem **(Muirhead)

If , where the arrays , , then for any array of positive numbers , we have

**Proof**

By Birkhoff’s theorem, we have

. Then by weighted AM-GM inequality,

Summing over all the possible , we have the desired inequality.

**References**

1. Albert W. Marshall and Ingram Olkin, *Inequalities: Theory of Majorization and Its Applications*. Academic Press, 1979.

2. Fedor Petrov, “Majorization Sequences”, http://www.mathlinks.ro/viewtopic.php?search_id=1697861374&t=5849

January 31, 2011 at 7:38 am |

This is a lovely exposition. Thanks for posting it. Cheers, M.

March 5, 2014 at 2:21 pm |

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