On the stability of Wright-convex functions
Let \(D\) be a convex subset of a linear space. A function \(f:D\to{\mathbb R}\) is called Wright-convex if \[ f\bigl(tx+(1-t)y\bigr)+f\bigl(ty+(1-t)x\bigr)\leq f(x)+f(y) \] for all \(x,y\in D\), \(t\in[0,1]\). The main result of the paper is the following Theorem: Let \(D\subset{\mathbb R}^N\) be a nonempty open and convex set and let \(\delta\geq 0\). Assume that a function \(f:D\to{\mathbb R}\) fulfils the inequality \[ f\bigl(tx+(1-t)y\bigr)+f\bigl(ty+(1-t)x\bigr)\leq f(x)+f(y)+2\delta \] for all \(x,y\in D\), \(t\in[0,1]\). Then there exists a Wright-convex function \(\varphi:D\to{\mathbb R}\) such that \[ \bigl|f(x)-\varphi(x)\bigr|\leq 2C_N\delta \] for all \(x\in D\), where \(C_N\) is a constant depending only on \(N\).
scientific article; zbMATH DE number 1945918