Schur convexity properties of the weighted arithmetic integral mean and Chebyshev functional

Authors

  • Long Bo-Yong Anhui University
  • Jiang Yue-Ping Hunan University
  • Chu Yu-Ming Huzhou Teachers College

Keywords:

weighted arithmetic integral mean, weighted Chebyshev functional, Schur convex, Schur geometrically convex, Schur harmonic convex

Abstract

In this paper, we discuss the Schur convexity, Schur geometrical convexity and Schur harmonic convexity of the weighted arithmetic integral mean and Chebyshev functional. Several sufficient conditions, and necessary and sufficient conditions are established.

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References

I. Schur, Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzunsber, Berlin. Math. Ges., 22 (1923), pp. 9-20.

A. W. Marshall and I. Olkin, Inequalities: Theory of majorization and its applications, Academic Press, New York, 1979.

W. D. Jiang, Some properties of dual form of the Hamy's symmetric function, J. Math. Inequal., 1(1) (2007), pp. 117-125, https://doi.org/10.7153/jmi-01-12

H. N. Shi, Y. M. Jiang and W. D. Jiang, Schur-convexity and Schur-geometrically concavity of Gini means, Comput. Math. Appl., 57(2) (2009), pp. 266-274, https://doi.org/10.1016/j.camwa.2008.11.001

K. Z. Guan, Some properties of a class of symmetric functions, J. Math. Anal. Appl., 336(1) (2007), pp. 70-80, https://doi.org/10.1016/j.jmaa.2007.02.064

K. Z. Guan, Schur-convexity of the complete symmetric function, Math. Inequal. Appl., 9(4) (2006), pp. 567-576, https://doi.org/10.7153/mia-09-52

K. Z. Guan and J. H. Shen, Schur-convexity for a class of symmetric function and its applications, Math. Inequal. Appl., 9(2) (2006), pp. 199-210, https://doi.org/10.7153/mia-09-20

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(2) (2006), pp. 219-224, https://doi.org/10.7153/mia-09-22

F. Qi, A note on Schur-convexity of extended mean values, Rocky Mountain J. Math., 35(5) (2005), pp. 1787-1793, https://doi.org/10.1216/rmjm/1181069663

F. Qi, J. Sándor, S. S. Dragomir and A. Sofo, Notes on the Schur-convexity of the extended mean values, Taiwanese J. Math., 9(3) (2005), pp. 411-420, https://doi.org/10.11650/twjm/1500407849

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(1) (2008), pp. 229-238, https://doi.org/10.1215/kjm/1250280982

K. Z. Guan, A class of symmetric functions for multiplicatively convex function, Math. Inequal. Appl., 10(4) (2007), pp. 745-753, https://doi.org/10.7153/mia-10-69

Y. M. Chu, X. M. Zhang and G. D. Wang, The Schur geometrical convexity of the extended mean values, J. Convex Anal., 15(4) (2008), pp. 707-718,

Y. M. Chu, G. D. Wang and X. M. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetric function, Math. Nachr., 284(5-6) (2011), pp. 653-663, https://doi.org/10.1002/mana.200810197

S. H. Wu, Generalization and sharpness of the power means inequality and their applications, J. Math. Anal. Appl., 312(2) (2005), pp. 637-652, https://doi.org/10.1016/j.jmaa.2005.03.050

X. M. Zhang and Y. M. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain J. Math., 40(3) (2010), pp. 1061-1068, https://doi.org/10.1216/rmj-2010-40-3-1061

N. Elezović and J. Pecarić, A note on Schur-convex function, Rocky Mountain J. Math., 30(3) (2000), pp. 853-856, https://doi.org/10.1216/rmjm/1021477248

X. M. Zhang and Y. M. Chu, The Schur geometrical convexity of integral arithmetic mean, Int. J. Pure Appl. Math., 41(7) (2007), pp. 919-925.

S. S. Dragomir and Th. M. Rassias, Ostrowski type inequalities and applicatons in numerical integration, Kluwer Academic, Dordrecht, 2002.

P. Cerone and S.S. Dragomir, New bounds for the Cebyšev functional, Appl. Math. Lett., 18(6) (2005), pp. 603-611, https://doi.org/10.1016/j.aml.2003.09.013

P. Cerone, On Chebyshev funtional bounds, Differential Difference Equations and Applications, Hindawi Publ. Corp., New York, 2006, pp. 267-277.

S. S. Dragomir, Some integral inequalities of Grüss type, Indian J. Pure Appl. Math., 31(4) (2002), pp. 397-415.

N. Elezović, Lj. Marangunić and J. Pecarić, Some improvements of Grüss type inequality, J. Math. Inequal., 1(3) (2007), pp. 425-436, https://doi.org/10.7153/jmi-01-36

F. Zafar and N. A. Mir, A note on the generalization of some new Čebyšev type inequlities, Tamsui Oxf. J. Inf. Math. Sci., 27(2) (2011), pp. 149-157.

V. Culjak, Schur-convexity of the weighted Cebyšev functional, J. Math. Inequal., 5(2) (2011), pp. 213-217, https://doi.org/10.7153/jmi-05-19

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Published

2013-02-01

How to Cite

Bo-Yong, L., Yue-Ping, J., & Yu-Ming, C. (2013). Schur convexity properties of the weighted arithmetic integral mean and Chebyshev functional. Rev. Anal. Numér. Théor. Approx., 42(1), 72–81. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2013-vol42-no1-art5

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