On approximately convex functions

Páles, Zsolt

doi:10.1090/S0002-9939-02-06552-8

On approximately convex functions
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Proc. Amer. Math. Soc. 131 (2003), 243-252 Request permission

Abstract:

A real valued function $f$ defined on a real interval $I$ is called $(\varepsilon ,\delta )$-convex if it satisfies \[ f(tx+(1-t)y)\le tf(x)+(1-t)f(y) + \varepsilon t(1-t)|x-y| + \delta \quad \text {for}\ x,y\in I, t\in [0,1]. \] The main results of the paper offer various characterizations for $(\varepsilon ,\delta )$-convexity. One of the main results states that $f$ is $(\varepsilon ,\delta )$-convex for some positive $\varepsilon$ and $\delta$ if and only if $f$ 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 $\varepsilon =0$, the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called $\delta$-convexity.

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