On Approximately Convex Functions

Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary

Key words and phrases: Convexity, $ (\eps,\delta)$-convexity, Stability of convexity, $ (\eps,\delta)$-subgradient, $ (\eps,\de)$-subdifferentiability

1991 Mathematics Subject Classification: Primary 26A51, 26B25

Abstract: 
A real valued function $ f$ defined on a real interval $ I$ is called $ (\eps,\de)$-convex if it satisfies 
$\displaystyle f(tx+(1-t)y)\le tf(x)+(1-t)f(y) + \eps t(1-t)\vert x-y\vert + \de $

for $ x,y\in I$, $ t\in[0,1]$.

The main results of the paper offer various characterizations for $ (\eps,\de)$-convexity. One of the main results states that $ f$ is $ (\eps,\delta)$-convex for some positive $ \eps$ 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 $ \eps=0$, the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so called $ \de$-convexity.