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The optimization for the inequalities of power means

Journal of Inequalities and Applications20062006:46782

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

©  J.Wen andW.-L.Wang. 2006

Received: 14 November 2005

Accepted: 14 July 2006

Published: 19 October 2006

Abstract

Let be the th 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’ Affiliations

(1)
Department of Mathematics and Computer Science, Chengdu University

References

  1. Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities, Mathematics and Its Applications (East European Series). Volume 31. D. Reidel, Dordrecht; 1988:xx+459.Google Scholar
  2. Department of Mathematics and Mechanics of Beijing University : Higher Algebra. People's Education Press, Beijing; 1978.Google Scholar
  3. Gardner RJ: The Brunn-Minkowski inequality. Bulletin of the American Mathematical Society. New Series 2002,39(3):355–405. 10.1090/S0273-0979-02-00941-2View ArticleMATHMathSciNetGoogle Scholar
  4. Guan K: Schur-convexity of the complete elementary symmetric function. Journal of Inequalities and Applications 2006, 2006: 9 pages.View ArticleMATHMathSciNetGoogle Scholar
  5. Kuang JC: Applied Inequalities. Hunan Education Press, Changsha; 2004.Google Scholar
  6. Lai L, Wen JJ: Generalization for Hardy's inequality of convex function. Journal of Southwest University for Nationalities (Natural Science Edition) 2003,29(3):269–274.Google Scholar
  7. Leng G, Zhao C, He B, Li X: Inequalities for polars of mixed projection bodies. Science in China. Series A. Mathematics 2004,47(2):175–186.MathSciNetView ArticleMATHGoogle Scholar
  8. Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974,81(8):879–883. 10.2307/2319447MathSciNetView ArticleMATHGoogle Scholar
  9. Liu Z: Comparison of some means. Journal of Mathematical Research and Exposition 2002,22(4):583–588.MathSciNetMATHGoogle Scholar
  10. Macdonald IG: Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs. 2nd edition. The Clarendon Press, Oxford University Press, New York; 1995:x+475.MATHGoogle Scholar
  11. Marshall AW, Olkin I: Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering. Volume 143. Academic Press, New York; 1979:xx+569.Google Scholar
  12. Minc H: Permanents. Addison-Wesley, Massachusetts; 1988.MATHGoogle Scholar
  13. Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series). Volume 61. Kluwer Academic, Dordrecht; 1993:xviii+740.View ArticleMATHGoogle Scholar
  14. Pečarić JE, Svrtan D: New refinements of the Jensen inequalities based on samples with repetitions. Journal of Mathematical Analysis and Applications 1998,222(2):365–373. 10.1006/jmaa.1997.5839MathSciNetView ArticleMATHGoogle Scholar
  15. Timofte V: On the positivity of symmetric polynomial functions. I. General results. Journal of Mathematical Analysis and Applications 2003,284(1):174–190. 10.1016/S0022-247X(03)00301-9MathSciNetView ArticleMATHGoogle Scholar
  16. Wang BY: An Introduction to the Theory of Majorizations. Beijing Normal University Press, Beijing; 1990.Google Scholar
  17. Wang ZL, Wang XH: Quadrature formula and analytic inequalities-on the separation of power means by logarithmic mean. Journal of Hangzhou University 1982,9(2):156–159.Google Scholar
  18. Wang W-L, Wang PF: A class of inequalities for the symmetric functions. Acta Mathematica Sinica 1984,27(4):485–497.MathSciNetMATHGoogle Scholar
  19. Wang W-L, Wen JJ, Shi HN: Optimal inequalities involving power means. Acta Mathematica Sinica 2004,47(6):1053–1062.MathSciNetMATHGoogle Scholar
  20. Wei Z, Qi L, Birge JR: A new method for nonsmooth convex optimization. Journal of Inequalities and Applications 1998,2(2):157–179. 10.1155/S1025583498000101MathSciNetMATHGoogle Scholar
  21. Wen JJ: The optimal generalization of A-G-H inequalities and its applications. Journal of Shaanxi Normal University 2004, 12–16.Google Scholar
  22. Wen JJ: Hardy means and their inequalities. to appear in Journal of Mathematics to appear in Journal of MathematicsGoogle Scholar
  23. Wen JJ, Wang W-L, Lu YJ: The method of descending dimension for establishing inequalities. Journal of Southwest University for Nationalities 2003,29(5):527–532.Google Scholar
  24. Wen JJ, Xiao CJ, Zhang RX: Chebyshev's inequality for a class of homogeneous and symmetric polynomials. Journal of Mathematics 2003,23(4):431–436.MathSciNetMATHGoogle Scholar
  25. Wen JJ, Zhang RX, Zhang Y: Inequalities involving the means of variance and their applications. Journal of Sichuan University. Natural Science Edition 2003,40(6):1011–1018.MathSciNetMATHGoogle Scholar
  26. Zheng WX, Wang SW: An Introduction to Real and Functional Analysis (no.2). People's Education Press, Shanghai; 1980.Google Scholar

Copyright

© J.Wen andW.-L.Wang. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.