Taiwanese Journal of Mathematics


Li-Li Fu, Bo-Yan Xi, and H. M. Srivastava

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In this paper, we give the sufficient as well as necessary condition of the Schur-convexity and Schur-harmonic-convexity of the generalized Heronian means with two positive numbers. Our main results provide the perfected versions of the results given in 2008 by Shi et al. [9].

Article information

Taiwanese J. Math., Volume 15, Number 6 (2011), 2721-2731.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 05E05: Symmetric functions and generalizations
Secondary: 26B25: Convexity, generalizations

Heronian means generalized Heronian means Schur-convexity Schur-harmonic-convexity arithmetic-geometric-harmonic means inequalities Schur-geometric-convexity


Fu, Li-Li; Xi, Bo-Yan; Srivastava, H. M. SCHUR-CONVEXITY OF THE GENERALIZED HERONIAN MEANS INVOLVING TWO POSITIVE NUMBERS. Taiwanese J. Math. 15 (2011), no. 6, 2721--2731. doi:10.11650/twjm/1500406493. https://projecteuclid.org/euclid.twjm/1500406493

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  • H. Alzer and W. Janous, Solution of problem 8*, Crux Math., 13 (1987), 173-178.
  • P. S. Bullen, D. S. Mitrinović and P. M. Vasić, Means and Their Inequalities, Kluwer Academic Publishers, Dordrecht, Boston and London, 1988.
  • K.-Z. Guan and H.-T. Zhu, The generalized Heronian mean and its inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 17 (2006), 60-75.
  • W. Janous, A note on generalized Heronian means. Math. Inequal. Appl., 4 (2001), 369-375.
  • G. Jia and J.-D. Cao, A new upper bound of the logarithmic mean, J. Inequal. Pure Appl. Math., 4 (2003), Article 80, 1-4 (electronic).
  • D.-M. Li, C. Gu and H.-N. Shi, Schur convexity of the power-type generalization of Heronian mean, Math. Practice and Theory, 36 (2006), 387-390 (in Chinese).
  • Q.-J. Mao, Dual means, logarithmic and Heronian dual means of two positivenumbers, J. Suzhou Coll. Ed., 16 (1999), 82-85 (in Chinese).
  • A. W. Marshall and I. Olkin, Inequalities$:$ Theory of Majorization and ItsApplications, Mathematics in Science and Engineering, Vol. 143, Academic Press, New York and London, 1979.
  • H.-N. Shi, M. Bencze, S.-H. Wu and D.-M. Li, Schur convexity of generalized Heronian means involving two parameters, J. Inequal. Appl., 2008 (2008), Article ID 879273, 1-9.
  • 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), 219-224.
  • W.-F. Xia and Y.-M. Chu, Schur-convexity for a class of symmetric functions and its applications. J. Inequal. Appl., (2009), Article ID 493759, 1-15.
  • Z.-H. Zhang and Y.-D. Wu, The generalized Heron mean and its dual form, Appl. Math. E-Notes, 5 (2005), 16-23 (electronic).
  • Z.-H. Zhang, Y.-D. Wu and H. M. Srivastava, Generalized Vandermonde determinants and mean values, Appl. Math. Comput., 202 (2008), 300-310.
  • Z.-H. Zhang, Y.-D. Wu and A.-P. Zhao, The properties of the generalized Heron means and its dual form, RGMIA Res. Rep. Collect., 7 (2004), Article 1.
  • N.-G. Zheng, Z.-H. Zhang and X.-M. Zhang, Schur-convexity of two types ofone-parameter mean values in $n$ variables, J. Inequal. Appl., 2007 (2007), Article ID 78175, 1-10.