# Approximately Convex Functions

• Donald H. Hyers
• George Isac
• Themistocles M. Rassias
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

## 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:
$$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right),$$
(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 ε-convex (where ε > 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.

## 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.

## Authors and Affiliations

• Donald H. Hyers
• George Isac
• 1
• Themistocles M. Rassias
• 2