Acta Mathematicae Applicatae Sinica

, Volume 20, Issue 1, pp 123–132 | Cite as

Inequalities of Hadamard Type for r-Convex Functions in Carnot Groups

Original Papers


For a Carnot group G, we establish the relationship between extended mean values and r-convex functions which is introduced in this paper, which is a class of inequalities of Hadamard type for r-convex function on G.


r-convex function extended mean values Carnot group inequality 

2000 MR Subject Classification

22E25 43A80 26D15 


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  1. 1.
    Danielli, N., Garofalo, N., Nhieu, D. Notions of convexity in Carnot groups. Comm. Anal. Geom., 11: 263–341 (2003)MATHMathSciNetGoogle Scholar
  2. 2.
    Folland, G.B., Stein, E.M. Hardy Space on Homogeneous groups. Princeton University Press, Princeton, NJ, 1982Google Scholar
  3. 3.
    Gill, P.M., Pearce, C.E.M., Pečarić, J. Hadamard’s inequality for r-convex functions. J. Math. Anal. Appl., 215: 461–470 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Lu, G.Z., Manfredi, J.J., Stroffolini, B. Convex functions on the Heisenberg group. Calc. Var. Partial Differential Equations, to appear.Google Scholar
  5. 5.
    Pansu, P. Métriques de Carnot-Carathéodory et quasii-sométries des espacec symétriques de rang un. Ann. Math., 129: 1–60 (1989)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Pearce, C.E.M., Pečarić, J. A continuous analogue and an extension of Radó’s formulae for convex and concave functions. Bull. Austral. Math. Soc., 53: 229–233 (1996)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Pearce, C.E.M., Pečarić, J., Šimić, V. Stolarsky means and Hadamard’s inequality. J. Math. Anal. Appl., 220: 99–109 (1998)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Stolarsky, K.B. Generalization of the logarithmic mean. Math. Mag., 48: 87–92 (1975)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Uhrin, B. Some remarks about the convolution of unimodal functions. Ann. Probab., 12: 640–645 (1984)MATHMathSciNetGoogle Scholar
  10. 10.
    Yang, G.S., H Wang, D.Y. Refinements of Hadamaard’s inequality for r-convex functions. Indian J. Pure appl. Math., 32: 1571–1579 (2001)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Applied MathematicsNanjing University of Science and TechnologyNanjing 210094China
  2. 2.Department of Applied MathematicsHunan Institute of Science and TechnologyYueyang 414000, HunanChina

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