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Ukrainian Mathematical Journal

, Volume 61, Issue 10, pp 1541–1555 | Cite as

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

  • Wei-Feng Xia
  • Yu-Ming Chu
Article
For x = (x 1, x 2, …, x n ) ∈ (0, 1 ] n and r ∈ { 1, 2, … , n}, a symmetric function F n (x, r) is defined by the relation
$$ {F_n}\left( {x,r} \right) = {F_n}\left( {{x_1},{x_2}, \ldots, {x_n};r} \right) = \sum\limits_{1{ \leqslant_1} < {i_2} \ldots {i_r} \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 - {x_{{i_j}}}}}{{{x_{{i_j}}}}}} }, $$
where i 1, i 2 ,… , i n are positive integers. This paper deals with the Schur convexity and Schur multiplicative convexity of F n (x, r). As applications, some inequalities are established by using the theory of majorization.

Keywords

Symmetric Function Isoperimetric Inequality Mathematical Induction Analytic Inequality Linear Experiment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • Wei-Feng Xia
    • 1
  • Yu-Ming Chu
    • 1
  1. 1.Huzhou Teachers CollegeHuzhouChina

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