This paper continues the study of the general convex functions. In this paper we extension our the former objects of the function in contact and of the function in circled contact. The following main result is proved: Let $J \subset R$ be an open interval and let $x_i,y_i \in J$ $(i=1,\dots,n)$ be real numbers such that fulfilling $x_i \geq \cdots \geq x_n$, $y_i \geq \cdots \geq y_n$. Then, a necessary and sufficient condition in order that $\sum_{i=1}^{n} f(x_i) \geq 2\sum_{i=1}^{n} f(y_i)-n max{f(a), f(b), g(f(a), f(b))}$ holds for every general convex function $f: J \to R$ which is in contact with function $g: f(J)^2 \to R$ and for arbitrary $a,b \in J$ ($a \leq x_i \leq b$ for $i=1,\dots,n$), is that $\sum_{i=1}^{k} y_i \leq \sum_{i=1}^{k} x_i$ $k=1, \dots, n-1$, $\sum_{i=1}^{n} y_i = \sum_{i=1}^{n} x_i$.