Abstract
In this paper, the generalized Heronian mean of n-tuple positive real valued function was defined which was the extension of the generalized Heronian mean of n-tuple positive real variable. By so-called Schur’s condition, the Schur-convexity and Schur-geometric convexity and Schur-harmonic convexity are studied for the generalized Heronian mean of n-tuple positive real valued function, and derive some results. Then the Schur-convexity also was discussed for its quotient, and as applications, a inequality is derived.
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Alzer, H., Janous, W.: Solution of Problem 8*. Crux. Math. 13, 173–178 (1987)
Bullen, P.S., Mitrinvić, D.S., Vasić, P.M.: Means and Their Inequalities. Kluwer Academic Publishers, Dordrecht (1988)
Janous, W.: A Note on Generalized Heronian Means. Math. Inequal. Appl. 4, 369–375 (2001)
Shi, H.-N., Jiang, Y.-M., Jiang, W.-D.: Schur-Convexity and Schur-Geometrically Con-cavity of Gini Mean. Comput. Math. Appl. 57, 266–274 (2009)
Guan, K.-Z., Zhu, H.T.: The Generalized Heronian Mean and Its Inequalities. Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. 17, 60–75 (2006)
Jia, G., Cao, J.: A New Upper Bound of the Logarithmic Mean. J. Inequal. Pure Appl. Math. 4(4) Art. 80 (2003)
Liu, Z.: Comparison of Some Means. J. Math. Res. Exp. 22(40), 583–588 (2002)
Yang, Z.-H.: ON the Homogeneous Functions With Two Parameters and Its Monotonicity. J. Inequal. Pure Appl. Math. 6(4) Art. 101 (2005)
Yang, Z.-H.: ON the Log-convexity of Two-parameter Homogeneous Functions. Math. Inequal. Appl. 10(3), 499–516 (2007)
Yang, Z.-H.: On the Monotonicity and Log-convexity of a Four-parameter Homogeneous Mean. J. Inequal. Appl. Art. ID 149286, 12 pages (2008)
Yang, Z.-H.: Some Monotonictiy Results for the Ratio of Two-parameter Symmetric Homogeneous Functions. Int. J. Math. Math. Sci. Art. ID 591382, 12 pages (2009)
Zhang, Z.-H., Wu, Y.-D.: The Generalized Heron Mean and Its Dual Form. Appl. Math. E-Notes 5, 16–23 (2005)
Zhang, Z.-H., Wu, Y.-D.: The Properties of the Generalized Heron Means and Its Dual Form. RGMIA Res. Rep. Col l. 7(2) Art. 1 (2004)
Shi, H.-N., Bencze, M., Wu, S.-H., Li, D.-M.: Schur-Convexity of Generalized Heronian Means Involving Two Parameters. Journal of Inequalities and Applications, Article ID 879273, 9 page (2008)
Guan, K.-z.: Schur-Convexity of Generalized Heron means. Journal of Heng Yang Normal University 6, 1–3 (2006)
Guan, K.-z.: Some Properties of a Generalized Hamy Symmetric Function and Its Applications. J. Math. Anal. 376, 494–505 (2011)
Wang, B.-Y.: Foundations of Majorization Inequalities. Beijing Normal Univ. Press, Beijing (1990)
Marshall, A.M., Olkin, I.: Inequalities: Theory of Majorization and Its Application. Academies Press, New York (1979)
Zhang, X.-M.: Schur-Convex Functions and Isoperimetric Inequalities. Proc. Amer. Math. Soc. 2, 461–470 (1998)
Xia, W.-F., Chu, Y.-M.: Schur-Convexity for a Class of Symmetric Functions and Its Applications. Journal of Inequalities and Applications, Article ID 493759, 15 pages (2009)
Shi, H.-N.: Schur-Geometric Convexity for Differences of Means. Applied Mathematics E-Notes 10, 275–284 (2010)
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Zhang, Ty., Ji, Ap. (2011). Schur-Convexity of Generalized Heronian Mean. In: Liu, C., Chang, J., Yang, A. (eds) Information Computing and Applications. ICICA 2011. Communications in Computer and Information Science, vol 244. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27452-7_4
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DOI: https://doi.org/10.1007/978-3-642-27452-7_4
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