Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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APPROXIMATE CONVEXITY WITH RESPECT TO INTEGRAL ARITHMETIC MEAN

  • Zoldak, Marek (Faculty of Mathematics and Natural Science University of Rzeszow)
  • Received : 2013.11.21
  • Published : 2014.11.30
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Abstract

Let (${\Omega}$, $\mathcal{S}$, ${\mu}$) be a probabilistic measure space, ${\varepsilon}{\in}\mathbb{R}$, ${\delta}{\geq}0$, p > 0 be given numbers and let $P{\subset}\mathbb{R}$ be an open interval. We consider a class of functions $f:P{\rightarrow}\mathbb{R}$, satisfying the inequality $$f(EX){\leq}E(f{\circ}X)+{\varepsilon}E({\mid}X-EX{\mid}^p)+{\delta}$$ for each $\mathcal{S}$-measurable simple function $X:{\Omega}{\rightarrow}P$. We show that if additionally the set of values of ${\mu}$ is equal to [0, 1] then $f:P{\rightarrow}\mathbb{R}$ satisfies the above condition if and only if $$f(tx+(1-t)y){\leq}tf(x)+(1-t)f(y)+{\varepsilon}[(1-t)^pt+t^p(1-t)]{\mid}x-y{\mid}^p+{\delta}$$ for $x,y{\in}P$, $t{\in}[0,1]$. We also prove some basic properties of such functions, e.g. the existence of subdifferentials, Hermite-Hadamard inequality.

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References

  1. J. Aczel, A generalization of the notion of convex functions, Norske Vid. Selsk. Forhdl. Trondheim 19 (1947), no. 24, 87-90.
  2. G. Aumann, Convexe Functionen und die Induktion bei Ungleichungen zwischen Mittelwerten, S.-B. Math. Natur. Abt. Bayer. Akad. Wiss. Munchen (1933), 403-415.
  3. D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821-828. https://doi.org/10.1090/S0002-9939-1952-0049962-5
  4. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, PWN-Uniwersytet Slaski, Warszawa - Krakow - Katowice, 1985.
  5. J. Matkowski, Convex and affine functions with respect to a mean and a characterization of the weighted quasi-arithmetic means, Real Anal. Exchange 29 (2003/04), 229-246.
  6. J. Matkowski and J. Ratz, Convex functions with respect to an arbitrary mean, General inequalities, 7 (Oberwolfach, 1995), 249-258, Internat. Ser. Numer. Math., 123, Birkhuser, Basel, 1997.
  7. N. Merentes and K. Nikodem, Remarks on strongly convex functions, Aequat. Math. 80 (2010), no. 1-2, 193-199. https://doi.org/10.1007/s00010-010-0043-0
  8. C. Niculescu and L. E. Persson, Convex Functions and Their Applications, CMS Books in Mathematics, Springer, 2006.
  9. K. Nikodem and Zs. Pales, Generalized convexity and separation theorems, J. Conv. Anal. 14 (2007), no. 2, 239-247.
  10. Zs. Pales, On approximately convex functions, Proc. Amer. Math. Soc. 131 (2003), no. 1, 243-252. https://doi.org/10.1090/S0002-9939-02-06552-8
  11. B. T. Poljak, Existence theorems and convergence of minimizing sequences for extremal problems with constrains, Dokl. Akad. Nauk SSSR 166 (1966), 287-290.
  12. A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973.
  13. W. Rudin, Real and Complex Analysis, McGraw-Hill, Inc., 1974.
  14. Ja. Tabor, Jo. Tabor, and M. Zoldak, Approximately convex functions on topological vector spaces, Publ. Math. Debrecen 77 (2010), no. 1-2, 115-123.
  15. Ja. Tabor, Jo. Tabor, and M. Zoldak, Strongly midquasiconvex functions, J. Conv. Anal. 20 (2013), no. 2, 531-543.
  16. T. Zgraja, Continuity of functions which are convex with respect to means, Publ. Math. Debrecen 63 (2003), no. 3, 401-411.