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Schur convexity for a class of symmetric functions

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Abstract

The Schur convexity and concavity of a class of symmetric functions are discussed, and an open problem proposed by Guan in “Some properties of a class of symmetric functions” is answered. As consequences, some inequalities are established by use of the theory of majorization.

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References

  1. Bullen P S. Handbook of Means and Their Inequalities. Dordrecht: Kluwer Academic Publishers Group, 2003

    MATH  Google Scholar 

  2. Chan N N. Schur-convexity for A-optimal designs. J Math Anal Appl, 1997, 122: 1–6

    Article  Google Scholar 

  3. Chu Y, Zhang X. Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave. J Math Kyoto Univ, 2008, 48: 229–238

    MATH  MathSciNet  Google Scholar 

  4. Constantine G M. Schur-convex functions on the spectra of graphs. Discrete Math, 1983, 45: 181–188

    Article  MATH  MathSciNet  Google Scholar 

  5. Forcina A, Giovagnoli A. Homogeneity indices and Schur-convex functions. Statistica (Bologna), 1982, 42: 529–542

    MATH  MathSciNet  Google Scholar 

  6. Guan K. Schur-convexity of the complete symmetric function. Math Inequal Appl, 2006, 9: 567–576

    MATH  MathSciNet  Google Scholar 

  7. Guan K. The Hamy symmetric function and its generalization. Math Inequal Appl, 2006, 9: 797–805

    MATH  MathSciNet  Google Scholar 

  8. Guan K. A class of symmetric functions for multiplicatively convex function. Math Inequal Appl, 2007, 10: 745–753

    MATH  MathSciNet  Google Scholar 

  9. Guan K. Some properties of a class of symmetric functions. J Math Anal Appl, 2007, 336: 70–80

    Article  MATH  MathSciNet  Google Scholar 

  10. Guan K, Shen J. Schur-convexity for a class of symmetric function and its applications. Math Inequal Appl, 2006, 9: 199–210

    MATH  MathSciNet  Google Scholar 

  11. Hwang F K, Rothblum U G. Partition-optimization with Schur convex sum objective functions. SIAM J Discrete Math, 2004/2005, 18: 512–524

    Article  MathSciNet  Google Scholar 

  12. Hwang F K, Rothblum U G, Shepp L. Monotone optimal multipartitions using Schur convexity with respect to partial orders. SIAM J Discrete Math, 1993, 6: 533–547

    Article  MATH  MathSciNet  Google Scholar 

  13. Jiang W. Some properties of dual form of the Hamy’s symmetric function. J Math Inequal, 2007, 1: 117–125

    MATH  MathSciNet  Google Scholar 

  14. Marshall A W, Olkin I. Inequalities: Theory of Majorization and Its Applications. New York: Academic Press, 1979

    MATH  Google Scholar 

  15. Merkle M. Convexity, Schur-convexity and bounds for the gamma function involving the digamma function. Rocky Mountain J Math, 1998, 28: 1053–1066

    Article  MATH  MathSciNet  Google Scholar 

  16. Mitrinović D S. Analytic Inequalities. New York: Springer-Verlag, 1970

    MATH  Google Scholar 

  17. Mitrinović D S, Pečarić J E, Volenec V. Recent Advances in Geometric Inequalities. Dordrecht: Kluwer Academic Publishers Group, 1989

    MATH  Google Scholar 

  18. Pečarić J E, Proschan F, Tong Y L. Convex Functions, Partial Orderings, and Statistical Applications. Boston: Academic Press, 1992

    MATH  Google Scholar 

  19. Qi F. A note on Schur-convexity of extended mean values. Rocky Mountain J Math, 2005, 35: 1787–1793

    Article  MATH  MathSciNet  Google Scholar 

  20. Qi F, Sándor J, Dragomir S S, et al. Notes on the Schur-convexity of the extended mean values. Taiwanese J Math, 2005, 9: 411–420

    MATH  MathSciNet  Google Scholar 

  21. Shaked M, Shanthikumar J G, Tong Y L. Parametric Schur convexity and arrangement monotonicity properties of partial sums. J Multivariate Anal, 1995, 53: 293–310

    Article  MATH  MathSciNet  Google Scholar 

  22. Shi H. Schur-convex functions related to Hadamard-type inequalities. J Math Inequal, 2007, 1: 127–136

    MATH  MathSciNet  Google Scholar 

  23. Shi H, Wu S, Qi F. An alternative note on the Schur-convexity of the extended mean values. Math Inequal Appl, 2006, 9: 219–224

    MATH  MathSciNet  Google Scholar 

  24. Stepniak C. Stochastic ordering and Schur-convex functions in comparison of linear experiments. Metrika, 1989, 36: 291–298

    Article  MATH  MathSciNet  Google Scholar 

  25. Stepniak C. An effective characterization of Schur-convex functions with applications. J Convex Anal, 2007, 14: 103–108

    MathSciNet  Google Scholar 

  26. Wu S. Generalization and sharpness of the power means inequality and their applications. J Math Anal Appl, 2005, 312: 637–652

    Article  MATH  MathSciNet  Google Scholar 

  27. Zhang X. Schur-convex functions and isoperimetric inequalities. Proc Amer Math Soc, 1998, 126: 461–470

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to YuMing Chu.

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Chu, Y., Xia, W. & Zhao, T. Schur convexity for a class of symmetric functions. Sci. China Math. 53, 465–474 (2010). https://doi.org/10.1007/s11425-009-0188-2

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  • DOI: https://doi.org/10.1007/s11425-009-0188-2

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