Schur-Convexity for a Class of Completely Symmetric Function Dual

Schur-Convexity for a Class of Completely Symmetric Function Dual

Authors

  • Huan-Nan Shi, Wei-Shih Du*

Keywords:

Schur-convexity, Schur-geometric convexity, Schur-harmonic convexity, completely symmetric function, dual form

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

A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Application (Second Edition),Springer, New York, 2011.

B. Y. Wang, Foundations of Majorization Inequalities, Beijing Normal University Press, Beijing, 1990. (in Chinese)

X. M. Zhang, Geometrically Convex Functions, An’hui University Press, Hefei, 2004. (in Chinese)

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.

K. Z. Guan, A class of symmetric functions for multiplicatively convex function, Mathematical Inequalities & Applications,2007, 10(4), 745-753.

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.

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.

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.

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.

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.

K.-Z. Guan, Schur-convexity of the complete symmetric function, Mathematical Inequalities & Applications, 2006, 9(4),567-576.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Ionel Rovent ¸a, A note on Schur-concave functions, Journal of Inequalities and Applications, 2012, 2012:159,doi:10.1186/1029-242X-2012-159.

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.

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.

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.

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.

X.-H. Zhang and Y.-M. Chu, New discussion to analytic Inequalities, Harbin, Harbin Institute of Technology Press, 2009.(inChinese)

H.-N. Shi. Majorization Theory and Analytical Inequalities, Harbin: Harbin Institute of Technology Press, 2012.(in Chinese)

H.-N. Shi. Schur-Convex Functions and Inequalities, Harbin: Harbin Institute of Technology Press, 2012.(in Chinese)

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Published

2019-12-31

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Articles

How to Cite

Schur-Convexity for a Class of Completely Symmetric Function Dual. (2019). Advances in the Theory of Nonlinear Analysis and Its Application, 3(2), 74-89. https://atnaea.org/index.php/journal/article/view/74