Approximately Convex Functions
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Abstract
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:
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 ε-convex (where ε > 0) if the inequality:
holds for all \(\lambda\)in [0, 1] and all x, y in S.
$$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right),$$
(8.1)
$$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right) + \varepsilon $$
(8.2)
Keywords
Functional Equation Convex Function Convex Subset Interior Point Topological Vector Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer Science+Business Media New York 1998