Volume 4,  Issue 2, 2003

Article 37



E-Mail: cej3@lehigh.edu 
URL: http://www.lehigh.edu/~cej3/cej3.html

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 $ n$-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 $ i$ such that $ x_i<y_i$; 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 $ F$ replaced by any symmetric polynomial with positive coefficients. Moreover, we find that the inequalities are not specific to expressions of the form $ x^y$, rather they hold for any function $ g(x,y)$ 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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