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: 8.1$$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right),$$where$$0 \leqslant \lambda \leqslant 1$$, with x and y in Rn, A function f : S→R, where S is a convex subset of Rn, will be called ε-convex (where ε > 0) if the inequality: 8.2$$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right) + \varepsilon $$ holds for all $$\lambda$$in [0, 1] and all x, y in S.