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On approximately convex functions |
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| Authors: | Zsolt Pá les |
| Institution: | Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary |
| Abstract: | A real valued function  defined on a real interval  is called  -convex if it satisfies ![\begin{displaymath}f(tx+(1-t)y)\le tf(x)+(1-t)f(y) + \varepsilon t(1-t)\vert x-y\vert + \delta \quad \text{for} x,y\in I,\, t\in0,1]. \end{displaymath}](http://www.ams.org/proc/2003-131-01/S0002-9939-02-06552-8/gif-abstract0/img4.gif){kind=small} The main results of the paper offer various characterizations for  -convexity. One of the main results states that  is  -convex for some positive  and  if and only if  can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case  , the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called  -convexity. |
| Keywords: | Convexity $(\varepsilon \delta)$-convexity stability of convexity $(\varepsilon \delta)$-subgradient $(\varepsilon \delta)$-subdifferentiability |
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