Characterizations of r-Convex Functions

  • Y. X. ZhaoEmail author
  • S. Y. Wang
  • L. Coladas Uria


This paper discusses some properties of r-convexity and its relations with some other types of convexity. A characterization of convex functions in terms of r-convexity is given without assuming differentiability. The concept of strict r-convexity is introduced. For a twice continuously differentiable function f, it is shown that the strict r-convexity of f is equivalent to a certain condition on 2 f. Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.


Strict r-convexity r-convexity Positive-semidefinite matrices Positive-definite matrices Convex functions Quasiconvex functions 


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  1. 1.
    Klinger, A., Mangasarian, O.L.: Logarithmic convexity and geometric programming. J. Math. Anal. Appl. 24, 388–408 (1968) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Antczak, T.: (p,r)-invex sets and functions. J. Math. Anal. Appl. 263, 355–379 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Luo, H.Z., Xu, Z.K.: Note on characterizations of prequasi-invex functions. J. Optim. Theory Appl. 120, 429–439 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hai, N.N., Phu, H.X.: Symmetrically γ-convex functions. Optimization 46, 1–23 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ponstein, J.: Seven kinds of convexity. SIAM Rev. 9, 115–119 (1967) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Avriel, M.: r-convex functions. Math. Program. 2, 309–323 (1972) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bullen, P.S.: A criterion for n-convexity. Pac. J. Math. 36, 81–98 (1971) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Phu, H.X.: Some properties of globally δ-convex functions. Optimization 35, 23–41 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Phu, H.X., Hai, N.N.: Some analytical properties of γ-convex functions on the real line. J. Optim. Theory Appl. 91, 671–694 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York (1973), p. 210 zbMATHGoogle Scholar
  11. 11.
    Mukherjee, R.N., Keddy, L.V.: Semicontinuity and quasiconvex functions. J. Optim. Theory Appl. 94, 715–720 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Yang, X.M., Yang, X.Q., Teo, K.L.: Characterizations and applications of prequasi-invex functions. J. Optim. Theory Appl. 110, 645–668 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Yang, X.M., Liu, S.Y.: Three kinds of generalized convexity. J. Optim. Theory Appl. 86, 501–513 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Yang, X.M.: Convexity of semicontinuous functions. Oper. Res. Soc. India 31, 309–317 (1994) zbMATHGoogle Scholar
  15. 15.
    Galewska, E., Galewski, M.: r-convex transformability in nonlinear programming problems. Comment. Math. Univ. Carol. 46, 555–565 (2005) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Avriel, M.: Solution of certain nonlinear programs involving r-convex functions. J. Optim. Theory Appl. 11, 159–174 (1973) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Avriel, M.: Nonlinear Programming: Analysis and Methods, pp. 163 and 173, Prentice-Hall, Englewood Cliffs (1976) (in Chinese) zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of Systems Science, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.Department of Statistics and Operations Research, Faculty of MathematicsSantiago de Compostela UniversitySantiago de CompostelaSpain

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