Inequalities for Symmetric Polynomials - Bucknell University

Inequalities for Symmetric Polynomials

Inequalities for Symmetric Polynomials

Curtis Greene October 24, 2009

Inequalities for Symmetric Polynomials Co-authors

This talk is based on "Inequalities for Symmetric Means", with Allison Cuttler, Mark Skandera (to appear in European Jour. Combinatorics). "Inequalities for Symmetric Functions of Degree 3", with Jeffrey Kroll, Jonathan Lima, Mark Skandera, and Rengyi Xu (to appear). Other work in progress.

Available on request, or at haverford.edu/math/cgreene.

Inequalities for Symmetric Polynomials Inequalities for Averages and Means

Classical examples (e.g., Hardy-Littlewood-Polya)

THE AGM INEQUALITY:

x1

+ x2

+??? n

+ xn

(x1x2 ? ? ? xn)1/n

x 0.

Inequalities for Symmetric Polynomials Inequalities for Averages and Means

Classical examples (e.g., Hardy-Littlewood-Polya)

THE AGM INEQUALITY:

x1

+ x2

+??? n

+ xn

(x1x2 ? ? ? xn)1/n

x 0.

NEWTON'S INEQUALITIES:

ek (x) ek (x) ek (1) ek (1)

ek-1(x) ek+1(x) ek-1(1) ek+1(1)

x 0

Inequalities for Symmetric Polynomials Inequalities for Averages and Means

Classical examples (e.g., Hardy-Littlewood-Polya)

THE AGM INEQUALITY:

x1

+ x2

+??? n

+ xn

(x1x2 ? ? ? xn)1/n

x 0.

NEWTON'S INEQUALITIES:

ek (x) ek (x) ek (1) ek (1)

ek-1(x) ek+1(x) ek-1(1) ek+1(1)

x 0

MUIRHEAD'S INEQUALITIES: If || = |?|, then

m(x) m(1)

m?(x) m?(1)

x 0

iff

? (majorization).

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download