## 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 , then ,
with strict inequality if there is an index such that ; and for , then,
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.

 Other papers in this issue Bounds for Linear Recurrences with Restricted Coefficients Kenneth S. Berenhaut and Robert Lund Inequalities Arising out of the Value Distribution of a Differential Monomial Indrajit Lahiri and Shyamali Dewan Generalized Quasi-Variational Inequalities and Duality Jacqueline Morgan and Maria Romaniello On an Open Problem of Bai-Ni Guo and Feng Qi Zivorad Tomovski and Kostadin Trencevski An Asymptotic Expansion Gabriel Mincu On an Open Problem Regarding an Integral Inequality S. Mazouzi and Feng Qi Some Convergence Results on Sinc Interpolation Mohamad S. Sababheh, Abdul-Majid Nasayr and Kamel Al-Khaled Rate of Convergence of Summation-Integral Type Operators with Derivatives Of Bounded Variation Vijay Gupta, Vipin Vasishtha and M. K. Gupta Additions to the Telyakovskii's class S L. Leindler A Criterion for $p$-Valently Starlikeness Muhammet Kamali Inequalities Related to Rearrangements of Powers and Symmetric Polynomials C. Joita and P. Stanica Predictor-Corrector Methods for Generalised General Multivalued Mixed Quasi Variational Inequalities Chao Feng Shi, San Yang Liu and Lian Jun Li The Inequalities $G\leq L\leq I\leq A$ in $n$ Variables Zhen-Gang Xiao and Zhi-Hua Zhang Reverse Inequalities on Chaotically Geometric Mean via Specht Ratio, II Masatoshi Fujii, Jadranka Micic, J.E. Pecaric and Yuki Seo Epi-differentiability and Optimality Conditions for an Extremal Problem Under Inclusion Constraints T. Amahroq, N. Gadhi and Pr H. Riahi Some Grüss Type Inequalities in Inner Product Spaces S.S. Dragomir Monotonicity Results for the Gamma Function Chao-Ping Chen and Feng Qi Some Special Subclasses of Carathéodory's or Starlike Functions and Related Coefficient Problems Philippos Koulorizosa and Nikolas Samaris On the Product of Divisors of $n$ and of $s(n)$ Florian Luca New Inequalities Between Elementary Symmetric Polynomials Todor P. Mitev