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. The main results show that for all $\varphi$-Jensen convex function $f:D\to\mathbb{ R}$, for all rational $\lambda\in[0,1]$ and $x,y\in D$, the following inequality holds \[ f(\lambda x+(1-\lambda)y) \leq \lambda f(x)+(1-\lambda)f(y)+\sum_{ k=0}^\infty \frac{ 1}{ 2^k}\varphi\big(\mbox{ dist}(2^k\lambda,\mathbb{ Z})\cdot(x-y)\big). \] The infinite series on the right hand side is always convergent, moreover, for all rational $\lambda\in[0,1]$, it can be evaluated as a finite sum.
Categories: math.CA