Approximately convex functions

So far we have discussed the stability of various functional equations. In the present section, we consider the stability of a well-known functional inequality, namely the inequality defining convex functions: $$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right),$$ (8.1) where\(0 \leqslant \lambda \leqslant 1\), with x and y in R n , A function f : S→R, where S is a convex subset of R n , will be called e-convex (where e > 0) if the inequality: $$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right) + \varepsilon $$ (8.2) holds for all \(\lambda\)in [0, 1] and all x, y in S.