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

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Ukrainian Mathematical Journal Aims and scope

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.

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Authors and Affiliations

  1. Huzhou Teachers College, Huzhou, China

    Wei-Feng Xia & Yu-Ming Chu

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  1. Wei-Feng Xia
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  2. Yu-Ming Chu
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 10, pp. 1306–1318, October, 2009.

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Xia, WF., Chu, YM. Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications. Ukr Math J 61, 1541–1555 (2009). https://doi.org/10.1007/s11253-010-0296-8

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  • DOI: https://doi.org/10.1007/s11253-010-0296-8

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