Estimates for approximately Jensen convex functions
Estimates for approximately Jensen convex functions
In this paper functions $f:D\to\mathbb{R}$ satisfying the inequality \[ f\Big(\frac{x+y}{2}\Big)\leq\frac12f(x)+\frac12f(y) +\varphi\Big(\frac{x-y}{2}\Big) \qquad(x,y\in D) \] are studied, where $D$ is a nonempty convex subset of a real linear space $X$ and $\varphi:\{\frac12(x-y) : x,y \in D\}\to\mathbb{R}$ is a so-called error function. In this situation $f$ is said to be $\varphi$-Jensen convex. …