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The optimization for the inequalities of power means

Journal of Inequalities and Applications volume 2006, Article number: 46782 (2006) Cite this article

Abstract

Let be theth power mean of a sequence of positive real numbers, where, and. In this paper, we will state the important background and meaning of the inequality; a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Chengdu University, Chengdu, Sichuan, 610106, China

    Jiajin Wen & Wan-Lan Wang

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  1. Jiajin Wen
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  2. Wan-Lan Wang
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Correspondence to Jiajin Wen.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Wen, J., Wang, WL. The optimization for the inequalities of power means. J Inequal Appl 2006, 46782 (2006). https://doi.org/10.1155/JIA/2006/46782

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  • DOI: https://doi.org/10.1155/JIA/2006/46782

Keywords

  • Real Number
  • Open Problem
  • Mathematical Analysis
  • Positive Real Number
  • Symmetrical Polynomial