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2012 | 10 | 3 | 1017-1041
Tytuł artykułu

Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities

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EN
Abstrakty
EN
The connection between the functional inequalities $$f\left( {\frac{{x + y}} {2}} \right) \leqslant \frac{{f\left( x \right) + f\left( y \right)}} {2} + \alpha _J \left( {x - y} \right), x,y \in D,$$ and $$\int_0^1 {f\left( {tx + \left( {1 - t} \right)y} \right)\rho \left( t \right)dt \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right) + \alpha _{\rm H} \left( {x - y} \right),} x,y \in D,$$ is investigated, where D is a convex subset of a linear space, f: D → ℝ, α H;α J: D-D → ℝ are even functions, λ ∈ [0; 1], and ρ: [0; 1] →ℝ+ is an integrable nonnegative function with ∫01 ρ(t) dt = 1.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
3
Strony
1017-1041
Opis fizyczny
Daty
wydano
2012-06-01
online
2012-03-24
Twórcy
autor
autor
Bibliografia
  • [1] Bernstein F., Doetsch G., Zur Theorie der konvexen Funktionen, Math. Ann., 1915, 76(4), 514–526 http://dx.doi.org/10.1007/BF01458222
  • [2] Dragomir S.S., Pearce C.E.M., Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000, preprint available at http://ajmaa.org/RGMIA/papers/monographs/Master.pdf
  • [3] Hadamard J., Étude sur les propriétés des fonctions entières et en particulier d’une fonction considéréé par Riemann, J. Math. Pures Appl., 1893, 58, 171–215
  • [4] Hyers D.H., Ulam S.M., Approximately convex functions, Proc. Amer. Math. Soc., 1952, 3, 821–828 http://dx.doi.org/10.1090/S0002-9939-1952-0049962-5
  • [5] Házy A., On approximate t-convexity, Math. Inequal. Appl., 2005, 8(3), 389–402
  • [6] Házy A., On stability of t-convexity, In: Proceedings of MicroCAD 2007 International Scientific Conference, G, Miskolci Egyetem, Miskolc, 2007, 23–28
  • [7] Házy A., Páles Zs., On approximately midconvex functions, Bull. London Math. Soc., 2004, 36(3), 339–350 http://dx.doi.org/10.1112/S0024609303002807
  • [8] Házy A., Páles Zs., On approximately t-convex functions, Publ. Math. Debrecen, 2005, 66(3–4), 489–501
  • [9] Házy A., Páles Zs., On a certain stability of the Hermite-Hadamard inequality, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465, 571–583 http://dx.doi.org/10.1098/rspa.2008.0291
  • [10] Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, Prace Naukowe Uniwersytetu Slaskiego w Katowicach, 489, Panstwowe Wydawnictwo Naukowe, Warsaw, 1985
  • [11] Makó J., Páles Zs., Approximate convexity of Takagi type functions, J. Math. Anal. Appl., 2010, 369(2), 545–554 http://dx.doi.org/10.1016/j.jmaa.2010.03.063
  • [12] Makó J., Páles Zs., On approximately convex Takagi type functions, Proc. Amer. Math. Soc. (in press)
  • [13] Mitrinovic D.S., Lackovic I.B., Hermite and convexity, Aequationes Math., 1985, 28, 229–232 http://dx.doi.org/10.1007/BF02189414
  • [14] Niculescu C.P., Persson L.-E., Old and new on the Hermite-Hadamard inequality, Real Anal. Exchange, 2003/04, 29(2), 663–685
  • [15] Niculescu C.P., Persson L.-E., Convex Functions and Their Applications, CMS Books Math./Ouvrages Math. SMC, 23, Springer, New York, 2006
  • [16] Nikodem K., Riedel T., Sahoo P.K., The stability problem of the Hermite-Hadamard inequality, Math. Inequal. Appl., 2007, 10(2), 359–363
  • [17] Tabor J., Tabor J., Generalized approximate midconvexity, Control Cybernet., 2009, 38(3), 655–669
  • [18] Tabor J., Tabor J., Takagi functions and approximate midconvexity, J. Math. Anal. Appl., 2009, 356(2), 729–737 http://dx.doi.org/10.1016/j.jmaa.2009.03.053
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0027-5
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