Inequalities Related to Rearrangements of Powers and Symmetric Polynomials

Volume 4, Issue 2, 2003

Article 37

INEQUALITIES RELATED TO REARRANGEMENTS OF POWERS AND SYMMETRIC POLYNOMIALS

DEPARTMENT OF MATHEMATICS,
LEHIGH UNIVERSITY,
BETHLEHEM, PA 18015, USA.
E-Mail: cej3@lehigh.edu URL: http://www.lehigh.edu/~cej3/cej3.html

AUBURN UNIVERSITY MONTGOMERY,
DEPARTMENT OF MATHEMATICS,
MONTGOMERY,
AL 36124-4023, USA.
E-Mail: pstanica@mail.aum.edu URL: http://sciences.aum.edu/~stanica

Received 29 April, 2003; Accepted 19 May, 2003.
Communicated by: C. Niculescu

ABSTRACT. In [2] the second author proposed to find a description (or examples) of real-valued

-variable functions satisfying the following two inequalities:

if $x_i\leq y_i, i=1,\ldots,n$ , then $F(x_1,\ldots,x_n)\leq F(y_1,\ldots,y_n)$ ,

with strict inequality if there is an index

such that

; and for $0<x_1<x_2<\cdots<x_n$ , then,

$\displaystyle F(x_1^{x_2},x_2^{x_3},\ldots,x_n^{x_1})\leq F(x_1^{x_1},x_2^{x_2},\cdots,x_n^{x_n}).$

In this short note we extend in a direction a result of [2] and we prove a theorem that provides a large class of examples satisfying the two inequalities, with

replaced by any symmetric polynomial with positive coefficients. Moreover, we find that the inequalities are not specific to expressions of the form

, rather they hold for any function

that satisfies some conditions. A simple consequence of this result is a theorem of Hardy, Littlewood and Polya [1].

[1] G. HARDY, J.E. LITTLEWOOD and G. PÓLYA, Inequalities, Cambridge Univ. Press, 2001.

[2] P. STANICA, Inequalities on linear functions and circular powers, J. Ineq. in Pure and Applied Math., 3(3) (2002), Art. 43.

Key words:
Symmetric Polynomials, Permutations, Inequalities.

2000 Mathematics Subject Classification:
05E05, 11C08, 26D05.

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