Abstract

A real valued function f ⁣:D→Rf\colon D\to \R defined on an open convex subset DD of a normed space XX is called rationally (h,d)(h,d)-convex if it satisfies f(tx+(1−t)y)≤h(t)f(x)+h(1−t)f(y)+d(x,y)f\left(tx + (1-t)y \right) \leq h(t) f(x) + h(1-t) f(y) + d(x,y) for all x,y∈Dx,y\in D and t∈Q∩[0,1]t\in \Q \cap [0,1], where d ⁣:X×X→Rd\colon X \times X \to \R and h:[0,1]→Rh:[0,1] \to \R are given functions. Our main result is of Bernstein-Doetsch type. Namely, we prove that if ff is locally bounded from above at a point of DD and rationally (h,d)(h,d)-convex then it is continuous and (h,d)(h,d)-convex.

Author information

Contact details are reproduced from the original publication and may be historical.

Pál Burai

Dept. of Applied Mathematics and Probability Theory, University of Debrecen, 4010 Debrecen Pf. 12, Hungary

burai@inf.unideb.hu

Attila Házy

Dept. of Applied Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary

matha@uni-miskolc.hu

Keywords

  • Convexity
  • approximate convexity
  • h-convexity
  • s-convexity
  • Bernstein-Doetsch theorem
  • regularity properties of generalized convex functions

Mathematics Subject Classification

26A51, 26B25, 39B62

References

  1. F. Bernstein, G. Doetsch: Zur Theorie der konvexen Funktionen, Math. Ann. 76 (1915) 514–526.https://doi.org/10.1007/BF01458222Google Scholar
  2. P. Burai, A. Házy, T. Juhász: Bernstein-Doetsch type results for s-convex functions, Publ. Math. 75(1–2) (2009) 23–31.https://doi.org/10.5486/PMD.2009.4414Google Scholar
  3. W. W. Breckner: Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen, Publ. Inst. Math., Nouv. Sér. 23 (1978) 13–20.Google Scholar
  4. W. W. Breckner, G. Orbán: Continuity Properties of Rationally s-Convex Mappings with Values in Ordered Topological Linear Space, "Babes-Bolyai" University, Kolozsvár (1978).Google Scholar
  5. A. Házy: On approximate t-convexity, Math. Inequal. Appl. 8(3) (2005) 389–402.https://doi.org/10.7153/mia-08-37Google Scholar
  6. A. Házy: On the stability of t-convex functions, Aequationes Math. 74 (2007) 210–218.https://doi.org/10.1007/s00010-007-2880-zGoogle Scholar
  7. A. Házy, Zs. Páles: Approximately midconvex functions, Bull. Lond. Math. Soc. 36 (2004) 339–350.https://doi.org/10.1112/S0024609303002807Google Scholar
  8. A. Házy, Zs. Páles: On approximately t-convex functions, Publ. Math. 66(3–4) (2005) 489–501.https://doi.org/10.5486/PMD.2005.3123Google Scholar
  9. H. Hudzik, L. Maligranda: Some remarks on si -convex functions, Aequationes Math. 48 (1994) 100–111.https://doi.org/10.1007/BF01837981Google Scholar
  10. D. H. Hyers, S. M. Ulam: Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952) 821–828.https://doi.org/10.1090/S0002-9939-1952-0049962-5Google Scholar
  11. M. Kuczma: An Introduction to the Theory of Functional Equations and Inequalities, Państwowe Wydawnictwo Naukowe, Uniwersytet Śląski, Warszawa (1985).Google Scholar
  12. D. T. Luc, H. V. Ngai, M. Théra: Approximate convex functions, J. Nonlinear Convex Anal. 1(2) (2000) 155–176.Google Scholar
  13. C. T. Ng, K. Nikodem: On approximately convex functions, Proc. Amer. Math. Soc. 118(1) (1993) 103–108.https://doi.org/10.1090/S0002-9939-1993-1159176-XGoogle Scholar
  14. Zs. Páles: Bernstein-Doetsch-type results for general functional inequalities, Rocz. Nauk.-Dydakt., Pr. Mat. 17 (2000) 197–206.Google Scholar
  15. Zs. Páles: On approximately convex functions, Proc. Amer. Math. Soc. 131 (2003) 243–252.https://doi.org/10.1090/S0002-9939-02-06552-8Google Scholar
  16. S. Piccard: Sur des ensembles parfaits, Mém. Univ. Neuchâtel 16, Secrétariat de l’Université, Neuchâtel (1942).Google Scholar
  17. M. Pycia: A direct proof of the s-Hölder continuity of Breckner s-convex functions, Aequationes Math. 61 (2001) 128–130.https://doi.org/10.1007/s000100050165Google Scholar
  18. H. Steinhaus: Sur les distances des points des ensembles de mesure positive, Fundam. Math. 1 (1920) 93-104.https://doi.org/10.4064/fm-1-1-93-104Google Scholar
  19. S. Varošanec: On h-convexity, J. Math. Anal. Appl. 32 (2007) 303–311.https://doi.org/10.1016/j.jmaa.2006.02.086Google Scholar

Suggested citation

P. Burai, A. Házy. “On Approximately h-Convex Functions.” Journal of Convex Analysis 18 (2011), No. 2, 447–454. https://doi.org/10.68381/jca18028

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press.