The connection between the functional inequalities and 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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0027-5,
author = {Judit Mak\'o and Zsolt P\'ales},
title = {Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1017-1041},
zbl = {1243.39021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0027-5}
}
Judit Makó; Zsolt Páles. Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities. Open Mathematics, Tome 10 (2012) pp. 1017-1041. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0027-5/
[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 | Zbl 0047.29505
[5] Házy A., On approximate t-convexity, Math. Inequal. Appl., 2005, 8(3), 389–402 | Zbl 1070.26010
[6] Házy A., On stability of t-convexity, In: Proceedings of MicroCAD 2007 International Scientific Conference, G, Miskolci Egyetem, Miskolc, 2007, 23–28 | Zbl 1130.26006
[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 | Zbl 1052.26014
[8] Házy A., Páles Zs., On approximately t-convex functions, Publ. Math. Debrecen, 2005, 66(3–4), 489–501 | Zbl 1082.26007
[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 | Zbl 1201.26003
[12] Makó J., Páles Zs., On approximately convex Takagi type functions, Proc. Amer. Math. Soc. (in press) | Zbl 1273.39022
[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 | Zbl 1100.26002
[16] Nikodem K., Riedel T., Sahoo P.K., The stability problem of the Hermite-Hadamard inequality, Math. Inequal. Appl., 2007, 10(2), 359–363 | Zbl 1117.39019
[17] Tabor J., Tabor J., Generalized approximate midconvexity, Control Cybernet., 2009, 38(3), 655–669 | Zbl 1301.52002
[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 | Zbl 1188.26008