In this paper functions \(f \colon D \to\mathbb{R}\) satisfying the inequality \( f\big(\frac{x+y}{2}\big)\leq\frac12f(x)+\frac12f(y) +\varphi\big(\frac{x-y}{2}\big) \quad(x,y\in D)\) are studied, where \(D\) is a nonempty convex subset of a real linear space \(X\) and \(\varphi \colon \{\frac12(x-y) : x,y \in D\}\to\mathbb{R}\) 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 \(f \colon D \to\mathbb{R}\) , for all rational \(\lambda\in[0,1]\) and \(x,y\in D\) , the following inequality holds \( f(\lambda x+(1-\lambda)y) \leq \lambda f(x)+(1-\lambda)f(y)+\sum_{k=0}^\infty \frac{1}{2^k}\varphi((2^k\lambda,\mathbb{Z})\cdot(x-y)).\) 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.
