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Some properties for a class of symmetric functions with applications

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Abstract

For x = (x 1, x 2, ..., x n ) ∈ ℝ n+ , the symmetric function ψ n (x, r) is defined by

$$\psi _n (x,r) = \psi _n \left( {x_1 ,x_2 , \cdots ,x_n ;r} \right) = \sum\limits_{1 \leqslant i_1 < i_2 \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} } ,$$

, where r = 1, 2, ..., n and i 1, i 2, ..., i n are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmoni convexity of ψ n (x, r) are discussed. As applications, some inequalities are established by use of the theory of majorization.

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Authors and Affiliations

  1. School of Teachers Education, Huzhou Teachers College, Huzhou, 313000, P. R. China

    Wei-Feng Xia & Yu-Ming Chu

  2. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, P. R. China

    Xiao-Hui Zhan & Gen-Di Wang

Authors
  1. Wei-Feng Xia
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  2. Xiao-Hui Zhan
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  3. Gen-Di Wang
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  4. Yu-Ming Chu
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Corresponding author

Correspondence to Wei-Feng Xia.

Additional information

This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Hunan Province under Grant 09JJ6003, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

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Xia, WF., Zhan, XH., Wang, GD. et al. Some properties for a class of symmetric functions with applications. Indian J Pure Appl Math 43, 227–249 (2012). https://doi.org/10.1007/s13226-012-0012-5

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  • DOI: https://doi.org/10.1007/s13226-012-0012-5

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