Estimates for approximately Jensen convex functions

In this paper functions $f:D\to \mathbb{R}$ satisfying the inequality $$f\left(\frac{x+y}{2}\right)\le \frac{1}{2}f(x)+\frac{1}{2}f(y)+\phi \left(\frac{x-y}{2}\right)(x,y\in D)$$ are studied, where $D$ is a nonempty convex subset of a real linear space $X$ and $\phi :\{\frac{1}{2}(x-y):x,y\in D\}\to \mathbb{R}$ is a so-called error function. In this situation $f$ is said to be $\phi$-Jensen convex. The main results show that for all $\phi$-Jensen convex function $f:D\to \mathbb{R}$, for all rational $\lambda \in [0,1]$ and $x,y\in D$, the following inequality holds $$f(\lambda x+(1-\lambda )y)\le \lambda f(x)+(1-\lambda )f(y)+\sum _{k=0}^{\mathrm{\infty }}\frac{1}{2^k}\phi (dist(2^k\lambda ,\mathbb{Z})\cdot (x-y))\mathrm{.}$$ 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.