Aequationes mathematicae

, Volume 80, Issue 1–2, pp 193–199 | Cite as

Remarks on strongly convex functions

  • Nelson Merentes
  • Kazimierz NikodemEmail author


Some properties of strongly convex functions are presented. A characterization of pairs of functions that can be separated by a strongly convex function and a Hyers–Ulam stability result for strongly convex functions are given. An integral Jensen-type inequality and a Hermite–Hadamard-type inequality for strongly convex functions are obtained. Finally, a relationship between strong convexity and generalized convexity in the sense of Beckenbach is shown.

Mathematics Subject Classification (2000)

Primary 26A51 Secondary 39B62 


Strongly convex functions generalized convex functions Jensen inequality Hermite–Hadamard inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baron K., Matkowski J., Nikodem K.: A sandwich with convexity. Math. Pannonica 5/1, 139–144 (1994)MathSciNetGoogle Scholar
  2. 2.
    Beckenbach E.F.: Generalized convex functions. Bull. Am. Math. Soc. 43, 363–371 (1937)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bessenyei M., Páles Zs.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6/3, 379–392 (2003)Google Scholar
  4. 4.
    Bessenyei M., Páles Zs.: Characterization of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9/1, 53–62 (2006)Google Scholar
  5. 5.
    Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University (2002) (
  6. 6.
    Hiriart-Urruty J.-B., Lemaréchal C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)zbMATHGoogle Scholar
  7. 7.
    Hyers D.H., Ulam S.M.: Approximately convex functions. Proc. Am. Math. Soc. 3, 821–828 (1952)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, 2nd edn. PWN, Uniwersytet Śla̧ski, Warszawa, Kraków, Katowice, 1985. Birkhäuser, Basel (2009)Google Scholar
  9. 9.
    Niculescu, C.P., Persson, L.-E.: Convex Functions and their Applications. A Contemporary Approach, CMS Books in Mathematics, vol. 23. Springer, New York (2006)Google Scholar
  10. 10.
    Nikodem K., Páles Zs.: Generalized convexity and separation theorems. J. Conv. Anal. 14/2, 239–247 (2007)Google Scholar
  11. 11.
    Polyak B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet Math. Dokl. 7, 72–75 (1966)Google Scholar
  12. 12.
    Roberts A.W., Varberg D.E.: Convex Functions. Academic Press, New York (1973)zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Escuela de MatemáticasUniversidad Central de VenezuelaCaracasVenezuela
  2. 2.Department of Mathematics and Computer ScienceUniversity of Bielsko-BiałaBielsko-BiałaPoland

Personalised recommendations