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SCHUR POWER CONVEXITY OF GINI MEANS

  • Yang, Zhen-Hang
  • Received : 2011.10.25
  • Published : 2013.03.31

Abstract

In this paper, the Schur convexity is generalized to Schur $f$-convexity, which contains the Schur geometrical convexity, harmonic convexity and so on. When $f$ : ${\mathbb{R}}_+{\rightarrow}{\mathbb{R}}$ is defined as $f(x)=(x^m-1)/m$ if $m{\neq}0$ and $f(x)$ = ln $x$ if $m=0$, the necessary and sufficient conditions for $f$-convexity (is called Schur $m$-power convexity) of Gini means are given, which generalize and unify certain known results.

Keywords

Schur convexity;Schur power convexity;Gini means

References

  1. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl. 335 (2007), no. 2, 1294-1308. https://doi.org/10.1016/j.jmaa.2007.02.016
  2. J. S. Aujla and F. C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003), 217-233. https://doi.org/10.1016/S0024-3795(02)00720-6
  3. Y. M. Chu and Y.-P. Lv, The Schur harmonic convexity of the Hamy symmetric function and its applications, J. Inequal. Appl. 2009 (2009), Art. ID 838529, 10 pages.
  4. Y. M. Chu and X. M. Zhang, Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave, J. Math. Kyoto Univ. 48 (2008), no. 1, 229-238. https://doi.org/10.1215/kjm/1250280982
  5. Y. M. Chu, X. M. Zhang, and G.-D. Wang, The Schur geometrical convexity of the extended mean values, J. Convex Anal. 15 (2008), no. 4, 707-718.
  6. Y. M. Chu and W. F. Xia, Solution of an open problem for Schur convexity or concavity of the Gini mean values, Sci. China Ser. A 52 (2009), no. 10, 2099-2106. https://doi.org/10.1007/s11425-009-0116-5
  7. G. M. Constantine, Schur convex functions on the spectra of graphs, Discrete Math. 45 (1983), no. 2-3, 181-188. https://doi.org/10.1016/0012-365X(83)90034-1
  8. P. Czinder and Zs. Pales, A general Minkowski-type inequality for two variable Gini means, Publ. Math. Debrecen 57 (2000), no. 1-2, 203-216.
  9. P. Czinder and Zs. Pales, Local monotonicity properties of two-variable Gini means and the comparison theorem revisited, J. Math. Anal. Appl. 301 (2005), no. 2, 427-438. https://doi.org/10.1016/j.jmaa.2004.08.006
  10. Z. Daroczy and L. Losonczi, Uber den Vergleich von Mittelwerten, Publ. Math. Debrecen 17 (1970), 289-297.
  11. D. Farnsworth and R. Orr, Gini means, Amer. Math. Monthly 93 (1986), no. 8, 603-607. https://doi.org/10.2307/2322316
  12. A. Forcina and A. Giovagnoli, Homogeneity indices and Schur-convex functions, Statistica 42 (1982), no. 4, 529-542.
  13. C. Gini, Diuna formula comprensiva delle media, Metron 13 (1938), 3-22.
  14. Ch. Gu and H. N. Shi, Schur-convexity and Schur-geometric convexity of Lehmer means, Math. Prac. Theory 39 (2009), no. 12, 183-188.
  15. G. H. Hardy, J. E. Littlewood, and G. Polya, Some simple inequalities satisfied by convex functions, Messenger Math. 58 (1929), 145-152.
  16. F. K. Hwang and U. G. Rothblum, Partition-optimization with Schur convex sum objective functions, SIAM J. Discrete Math. 18 (2004), no. 3, 512-524. https://doi.org/10.1137/S0895480198347167
  17. F. K. Hwang, U. G. Rothblum, and L. Shepp, Monotone optimal multipartitions using Schur convexity with respect to partial orders, SIAM J. Discrete Math. 6 (1993), no. 4, 533-547. https://doi.org/10.1137/0406042
  18. D.-M. Li, Ch. Gu, and H.-N. Shi, Schur convexity of the power-type generalization of Heronian mean, Math. Prac. Theory 36 (2006), no. 9, 387-390.
  19. D.-M. Li and H.-N. Shi, Schur convexity and Schur-geometrically concavity of generalized exponent mean, J. Math. Inequal. 3 (2009), no. 2, 217-225.
  20. Zh. Liu, Minkowski's inequality for extended mean values, Proceedings of the Second ISAAC Congress, Vol. 1 (Fukuoka, 1999), 585-592, Int. Soc. Anal. Appl. Comput. 7, Kluwer Acad. Publ., Dordrecht, 2000.
  21. L. Losonczi, Inequalities for integral mean values, J. Math. Anal. Appl. 61 (1977), no. 3, 586-606. https://doi.org/10.1016/0022-247X(77)90164-0
  22. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, New York, Academic Press, 1979.
  23. M. Merkle, Convexity, Schur-convexity and bounds for the gamma function involving the digamma function, Rocky Mountain J. Math. 28 (1998), no. 3, 1053-1066. https://doi.org/10.1216/rmjm/1181071755
  24. C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl. 3 (2000), no. 2, 155-167.
  25. E. Neuman and J. Sandor, Inequalities involving Stolarsky and Gini means, Math. Pannon. 14 (2003), no. 1, 29-44.
  26. E. Neuman and Zs. Pales, On comparison of Stolarsky and Gini means, J. Math. Anal. Appl. 278 (2003), no. 2, 274-284. https://doi.org/10.1016/S0022-247X(02)00319-0
  27. Zs. Pales, Comparison of two variable homogeneous means, General inequalities, 6 (Oberwolfach, 1990), 59-70, Internat. Ser. Numer. Math., 103, Birkhauser, Basel, 1992.
  28. F. Qi, A note on Schur-convexity of extended mean values, Rocky Mountain J. Math. 35 (2005), no. 5, 1787-1793. https://doi.org/10.1216/rmjm/1181069663
  29. F. Qi, J. Sandor, and S. S. Dragomir, Notes on the Schur-convexity of the extended mean values, Taiwanese J. Math. 9 (2005), no. 3, 411-420.
  30. J. Sandor, A note on the Gini means, Gen. Math. 12 (2004), no. 4, 17-21.
  31. J. Sandor, The Schur-convexity of Stolarsky and Gini means, Banach J. Math. Anal. 1 (2007), no. 2, 212-215. https://doi.org/10.15352/bjma/1240336218
  32. M. Shaked, J. G. Shanthikumar, and Y. L. Tong, Parametric Schur convexity and arrangement monotonicity properties of partial sums, J. Multivariate Anal. 53 (1995), no. 2, 293-310. https://doi.org/10.1006/jmva.1995.1038
  33. H. N. Shi, S. H. Wu, and F. Qi, An alternative note on the Schur-convexity of the extended mean values, Math. Inequal. Appl. 9 (2006), no. 2, 219-224.
  34. H.-N. Shi, Y.-M. Jiang, and W.-D. Jiang, Schur-convexity and Schur-geometrically concavity of Gini means, Comput. Math. Appl. 57 (2009), no. 2, 266-274. https://doi.org/10.1016/j.camwa.2008.11.001
  35. C. Stepniak, Stochastic ordering and Schur-convex functions in comparison of linear experiments, Metrika 36 (1989), no. 5, 291-298. https://doi.org/10.1007/BF02614102
  36. S. Toader and G. Toader, Complementaries of Greek means with respect to Gini means, Int. J. Appl. Math. Stat. 11 (2007), no. 7, 187-192.
  37. B.-Y. Wang, Foundations of Majorization Inequalities, Beijing Normal Univ. Press, Beijing, China, 1990.
  38. Z.-H. Wang, The necessary and sufficient condition for S-convexity and S-geometrically convexity of Gini mean, J. Beijing Ins. Edu. (Natural Science) 2 (2007), no. 5, 1-3.
  39. Z.-H. Wang and X.-M. Zhang, Necessary and sufficient conditions for Schur convexity and Schur-geometrically convexity of Gini means, Communications of inequalities researching 14 (2007), no. 2, 193-197.
  40. W.-F. Xia, The Schur harmonic convexity of Lehmer means, Int. Math. Forum 4 (2009), no. 41, 2009-2015.
  41. W.-F. Xia and Y.-M. Chu, Schur-convexity for a class of symmetric functions and its applications, J. Inequal. Appl. 2009 (2009), Art. ID 493759, 15 pages.
  42. Zh.-H. Yang, Simple discriminances of convexity of homogeneous functions and applications, Gaodeng Shuxue Yanjiu (Study in College Mathematics) 4 (2004), no. 7, 14-19.
  43. Zh.-H. Yang, On the homogeneous functions with two parameters and its monotonicity, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 101.
  44. Zh.-H. Yang, On the log-convexity of two-parameter homogeneous functions, Math. Inequal. Appl. 10 (2007), no. 3, 499-516.
  45. Zh.-H. Yang, On the monotonicity and log-convexity of a four-parameter homogeneous mean, J. Inequal. Appl. 2008 (2008), Art. ID 149286, 12 pages.
  46. Zh.-H. Yang, Some monotonictiy results for the ratio of two-parameter symmetric homogeneous functions, Int. J. Math. Math. Sci. 2009 (2009), Art. ID 591382, 12 pages.
  47. Zh.-H. Yang, Necessary and sufficient conditions for Schur convexity of the two-parameter symmetric homogeneous means, Appl. Math. Sci. (Ruse) 5 (2011), no. 64, 3183-3190.
  48. Zh.-H. Yang, The log-convexity of another class of one-parameter means and its applications, Bull. Korean Math. Soc. 49 (2012), no. 1, 33-47. https://doi.org/10.4134/BKMS.2012.49.1.033
  49. X.-M. Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc. 126 (1998), no. 2, 461-470. https://doi.org/10.1090/S0002-9939-98-04151-3
  50. X.-M. Zhang, Geometrically Convex Functions, Hefei, An'hui University Press, 2004.

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