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Schur-Convexity for a Class of Completely Symmetric Function Dual

Year 2019, Volume: 3 Issue: 2, 74 - 89, 30.06.2019
https://doi.org/10.31197/atnaa.573926

Abstract

By using the decision theorem and properties of the Schur-convex function, the Schur-geometric convex function and the Schur-harmonic function, the Schur- convexity, Schur-geometric convexity and Schur-harmonic convexity of a class of complete symmetric functions are studied. As applications, some symmetric function inequalities are established.

References

  • [1] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Application (Second Edition),Springer, New York, 2011.
  • [2] B. Y. Wang, Foundations of Majorization Inequalities, Beijing Normal University Press, Beijing, 1990. (in Chinese)
  • [3] X. M. Zhang, Geometrically Convex Functions, An’hui University Press, Hefei, 2004. (in Chinese)
  • [4] Y. M. Chu, X. M. Zhang, and G. D. Wang, The Schur geometrical convexity of the extended mean values, Journal ofConvex Analysis, 2008, 15(4), 707-718.
  • [5] K. Z. Guan, A class of symmetric functions for multiplicatively convex function, Mathematical Inequalities & Applications,2007, 10(4), 745-753.
  • [6] T.-C. Sun, Y.-P. Lv, and Y.-M. Chu, Schur multiplicative and harmonic convexities of generalized Heronian mean in nvariables and their applications, International Journal of Pure and Applied Mathematics, 2009, 55(1), 25-33.
  • [7] Y. M. Chu, and T. C. Sun, The Schur harmonic convexity for a class of symmetric functions, Acta Mathematica Scientia,2010, 30B(5), 1501-1506.
  • [8] Y.-M. Chu, G.-D.Wang, and X.-H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetricfunction, Mathematische Nachrichten, 2011, 284(5-6), 653-663.
  • [9] Y.-M. Chu, and Y.-P. Lv, The Schur harmonic convexity of the Hamy symmetric function and its applications, Journal ofInequalities and Applications, 2009, Article ID 838529, 10 pages.
  • [10] W. F. Xia, and Y. M. Chu, Schur-convexity for a class of symmetric functions and its applications, Journal of Inequalitiesand Applications, 2009, Article ID 493759, 15 pages.
  • [11] K.-Z. Guan, Schur-convexity of the complete symmetric function, Mathematical Inequalities & Applications, 2006, 9(4),567-576.
  • [12] M. B. Sun, N. B. Chen, and S. H. Li, Some properties of a class of symmetric functions and its applications, MathematischeNachrichten, 2014, doi: 10.1002/mana.201300073.
  • [13] W.-F. Xia, and Y.-M. Chu, Schur convexity and Schur multiplicative convexity for a class of symmeric functions withapplications, Ukrainian Mathematical Journal, 2009, 61(10), 1541-1555.
  • [14] Ionel Rovent ¸a, Schur convexity of a class of symmetric functions, Annals of the University of Craiova, Mathematics andComputer Science Series, 2010, 37(1), 12-18.
  • [15] W.-F. Xia, and Y.-M. Chu, On Schur convexity of some symmetric functions, Journal of Inequalities and Applications,2010, Article ID 543250, 12 pages.
  • [16] J.-X. Meng, Y.-M. Chu, and X.-M. Tang, The Schur-harmonic-convexity of dual form of the Hamy symmetric function,Matematiqki Vesnik, 2010, 62(1), 37-46.
  • [17] Y.-M. Chu, W.-F. Xia, and T.-H. Zhao, Some properties for a class of symmetric functions and applications, Journal ofMathematical Inequalities, 2011, 5(1), 1-11.
  • [18] K.-Z. Guan, and R.-K. Guan, Some properties of a generalized Hamy symmetric function and its applications, Journal ofMathematical Analysis and Applications, 2011, 376, 494-505.
  • [19] W.-M. Qian, Schur convexity for the ratios of the Hamy and generalized Hamy symmetric functions, Journal of Inequalitiesand Applications, 2011, 2011:131, doi:10.1186/1029-242X-2011-131.
  • [20] Y.-M. Chu, W.-F. Xia, and X.-H. Zhang, The Schur concavity, Schur multiplicative and harmonic convexities of the seconddual form of the Hamy symmetric function with applications, Journal of Multivariate Analysis, 2012, 105(1), 412-421.
  • [21] Ionel Rovent ¸a, A note on Schur-concave functions, Journal of Inequalities and Applications, 2012, 2012:159,doi:10.1186/1029-242X-2012-159.
  • [22] W.-F. Xia, X.-H. Zhang, G.-D. Wang and Y.-M. Chu, Some properties for a class of symmetric functions with applications,Indian J. Pure Appl. Math., 2012, 43(3), 227-249.
  • [23] H.-N. Shi and J. Zhang, Schur-convexity of dual form of some symmetric functions, Journal of Inequalities and Applications,2013, 2013,295, doi:10.1186/1029-242X-2013-295.
  • [24] K.-S. Zhang, and H.-N. Shi, Schur convexity of dual form of the complete symmetric function, Mathematical Inequalities& Applications, 2013, 16(4), 963-970.
  • [25] H.-N. Shi, J. Zhang and Q.-H. Ma. Schur-convexity, Schur-geometric and Schur-harmonic convexity for a composite functionof complete symmetric function, SpringerPlus (2016) 5,296.
  • [26] X.-H. Zhang and Y.-M. Chu, New discussion to analytic Inequalities, Harbin, Harbin Institute of Technology Press, 2009.(inChinese)
  • [27] H.-N. Shi. Majorization Theory and Analytical Inequalities, Harbin: Harbin Institute of Technology Press, 2012.(in Chinese)
  • [28] H.-N. Shi. Schur-Convex Functions and Inequalities, Harbin: Harbin Institute of Technology Press, 2012.(in Chinese)
Year 2019, Volume: 3 Issue: 2, 74 - 89, 30.06.2019
https://doi.org/10.31197/atnaa.573926

Abstract

References

  • [1] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Application (Second Edition),Springer, New York, 2011.
  • [2] B. Y. Wang, Foundations of Majorization Inequalities, Beijing Normal University Press, Beijing, 1990. (in Chinese)
  • [3] X. M. Zhang, Geometrically Convex Functions, An’hui University Press, Hefei, 2004. (in Chinese)
  • [4] Y. M. Chu, X. M. Zhang, and G. D. Wang, The Schur geometrical convexity of the extended mean values, Journal ofConvex Analysis, 2008, 15(4), 707-718.
  • [5] K. Z. Guan, A class of symmetric functions for multiplicatively convex function, Mathematical Inequalities & Applications,2007, 10(4), 745-753.
  • [6] T.-C. Sun, Y.-P. Lv, and Y.-M. Chu, Schur multiplicative and harmonic convexities of generalized Heronian mean in nvariables and their applications, International Journal of Pure and Applied Mathematics, 2009, 55(1), 25-33.
  • [7] Y. M. Chu, and T. C. Sun, The Schur harmonic convexity for a class of symmetric functions, Acta Mathematica Scientia,2010, 30B(5), 1501-1506.
  • [8] Y.-M. Chu, G.-D.Wang, and X.-H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetricfunction, Mathematische Nachrichten, 2011, 284(5-6), 653-663.
  • [9] Y.-M. Chu, and Y.-P. Lv, The Schur harmonic convexity of the Hamy symmetric function and its applications, Journal ofInequalities and Applications, 2009, Article ID 838529, 10 pages.
  • [10] W. F. Xia, and Y. M. Chu, Schur-convexity for a class of symmetric functions and its applications, Journal of Inequalitiesand Applications, 2009, Article ID 493759, 15 pages.
  • [11] K.-Z. Guan, Schur-convexity of the complete symmetric function, Mathematical Inequalities & Applications, 2006, 9(4),567-576.
  • [12] M. B. Sun, N. B. Chen, and S. H. Li, Some properties of a class of symmetric functions and its applications, MathematischeNachrichten, 2014, doi: 10.1002/mana.201300073.
  • [13] W.-F. Xia, and Y.-M. Chu, Schur convexity and Schur multiplicative convexity for a class of symmeric functions withapplications, Ukrainian Mathematical Journal, 2009, 61(10), 1541-1555.
  • [14] Ionel Rovent ¸a, Schur convexity of a class of symmetric functions, Annals of the University of Craiova, Mathematics andComputer Science Series, 2010, 37(1), 12-18.
  • [15] W.-F. Xia, and Y.-M. Chu, On Schur convexity of some symmetric functions, Journal of Inequalities and Applications,2010, Article ID 543250, 12 pages.
  • [16] J.-X. Meng, Y.-M. Chu, and X.-M. Tang, The Schur-harmonic-convexity of dual form of the Hamy symmetric function,Matematiqki Vesnik, 2010, 62(1), 37-46.
  • [17] Y.-M. Chu, W.-F. Xia, and T.-H. Zhao, Some properties for a class of symmetric functions and applications, Journal ofMathematical Inequalities, 2011, 5(1), 1-11.
  • [18] K.-Z. Guan, and R.-K. Guan, Some properties of a generalized Hamy symmetric function and its applications, Journal ofMathematical Analysis and Applications, 2011, 376, 494-505.
  • [19] W.-M. Qian, Schur convexity for the ratios of the Hamy and generalized Hamy symmetric functions, Journal of Inequalitiesand Applications, 2011, 2011:131, doi:10.1186/1029-242X-2011-131.
  • [20] Y.-M. Chu, W.-F. Xia, and X.-H. Zhang, The Schur concavity, Schur multiplicative and harmonic convexities of the seconddual form of the Hamy symmetric function with applications, Journal of Multivariate Analysis, 2012, 105(1), 412-421.
  • [21] Ionel Rovent ¸a, A note on Schur-concave functions, Journal of Inequalities and Applications, 2012, 2012:159,doi:10.1186/1029-242X-2012-159.
  • [22] W.-F. Xia, X.-H. Zhang, G.-D. Wang and Y.-M. Chu, Some properties for a class of symmetric functions with applications,Indian J. Pure Appl. Math., 2012, 43(3), 227-249.
  • [23] H.-N. Shi and J. Zhang, Schur-convexity of dual form of some symmetric functions, Journal of Inequalities and Applications,2013, 2013,295, doi:10.1186/1029-242X-2013-295.
  • [24] K.-S. Zhang, and H.-N. Shi, Schur convexity of dual form of the complete symmetric function, Mathematical Inequalities& Applications, 2013, 16(4), 963-970.
  • [25] H.-N. Shi, J. Zhang and Q.-H. Ma. Schur-convexity, Schur-geometric and Schur-harmonic convexity for a composite functionof complete symmetric function, SpringerPlus (2016) 5,296.
  • [26] X.-H. Zhang and Y.-M. Chu, New discussion to analytic Inequalities, Harbin, Harbin Institute of Technology Press, 2009.(inChinese)
  • [27] H.-N. Shi. Majorization Theory and Analytical Inequalities, Harbin: Harbin Institute of Technology Press, 2012.(in Chinese)
  • [28] H.-N. Shi. Schur-Convex Functions and Inequalities, Harbin: Harbin Institute of Technology Press, 2012.(in Chinese)
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Huan-Nan Shi This is me

Wei-Shih Du

Publication Date June 30, 2019
Published in Issue Year 2019 Volume: 3 Issue: 2

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