AIMS Mathematics, 2021, 6(1): 296-303. doi: 10.3934/math.2021018

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Necessary and sufficient conditions on the Schur convexity of a bivariate mean

1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, China
2 College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China
3 School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China

In the paper, the authors find and apply necessary and sufficient conditions for a bivariate mean of two positive numbers with three parameters to be Schur convex or Schur harmonically convex respectively.
  Article Metrics


1. Y. M. Chu, G. D. Wang, X. H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetric function, Math. Nachr., 284 (2011), 653-663.

2. C. R. Fu, D. S. Wang, H. N. Shi, Schur-convexity for a mean of two variables with three parameters, Filomat, 32 (2018), 6643-6651.

3. L. L. Fu, B. Y. Xi, H. M. Srivastava, Schur-convexity of the generalized Heronian means involving two positive numbers, Taiwanese J. Math., 15 (2011), 2721-2731.

4. J. C. Kuang, Applied inequalities (Chang Yong Bu Deng Shi), 4 Eds., Shandong Press of Science and Technology, Ji'nan, China, 2010. (Chinese)

5. A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: Theory of majorization and its applications, 2 Eds., Springer Verlag, New York/Dordrecht/Heidelberg/London, 2011.

6. F. Qi, A note on Schur-convexity of extended mean values, Rocky Mountain J. Math., 35 (2005), 1787-1793.

7. F. Qi, J. Sándor, S. S. Dragomir, A. Sofo, Notes on the Schur-convexity of the extended mean values, Taiwanese J. Math., 9 (2005), 411-420.

8. F. Qi, X. T. Shi, M. Mahmoud, F. F. Liu, Schur-convexity of the Catalan-Qi function related to the Catalan numbers, Tbilisi Math. J., 9 (2016), 141-150.

9. H. N. Shi, Y. M. Jiang, W. D. Jiang, Schur-convexity and Schur-geometrically concavity of Gini mean, Comput. Math. Appl., 57 (2009), 266-274.

10. H. N. Shi, B. Mihaly, S. H. Wu, D. M. Li, Schur convexity of generalized Heronian means involving two parameters, J. Inequal. Appl., 2008 (2009). Available from:

11. H. N. Shi, S. H. Wu, F. Qi, An alternative note on the Schur-convexity of the extended mean values, Math. Inequal. Appl., 9 (2006), 219-224.

12. J. Sun, Z. L. Sun, B. Y. Xi, F. Qi, Schur-geometric and Schur-harmonic convexity of an integral mean for convex functions, Turkish J. Anal. Number Theory, 3 (2015), 87-89.

13. B. Y. Wang, Foundations of majorization inequalities, Beijing Normal Univ. Press, Beijing, China, 1990. (Chinese)

14. Y. Wu, F. Qi, Schur-harmonic convexity for differences of some means, Analysis (Munich), 32 (2012), 263-270.

15. Y. Wu, F. Qi, H. N. Shi, Schur-harmonic convexity for differences of some special means in two variables, J. Math. Inequal., 8 (2014), 321-330.

16. B. Y. Xi, D. D. Gao, T. Zhang, B. N. Guo, F. Qi, Shannon type inequalities for Kapur's entropy, Mathematics, 7 (2019). Available from:

17. B. Y. Xi, Y. Wu, H. N. Shi, F. Qi, Generalizations of several inequalities related to multivariate geometric means, Mathematics, 7 (2019). Available from:

18. W. F. Xia, Y. M. Chu, The Schur convexity of Gini mean values in the sense of harmonic mean, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 1103-1112.

19. H. P. Yin, H. N. Shi, F. Qi, On Schur m-power convexity for ratios of some means, J. Math. Inequal., 9 (2015), 145-153.

© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved