The following main result is proved: Let $J\subset\mathbb{R}$ be an open interval and let $x_{i},y_{i}\in J$ $(i=1,\ldots,n)$ be real numbers such that fulfilling $x_{1}\geq\cdots\geq x_{n},\qquad y_{1}\geq\cdots\geq y_{n}$. Then, a necessary and sufficient condition in order that $\sum_{i=1}^{n}f(x_{i})\geq 2\sum_{i=1}^{n}f(y_{i})-n\max\bigl\{f(a),f(b),g(f(a),f(b))\bigr\}$ holds for every general convex function $f: J\to \mathbb{R}$ which is in contact with function $g: f(J)^{2}\to\mathbb{R}$ and for arbitrary $a,b\in J$ ($a\geq x_{i}\geq b$ for $i=1,\ldots,n$), is that $\sum_{i=1}^{k}y_{i}\leq\sum_{i=1}^{k}x_{i}\quad (k=1,\ldots,n-1),\qquad \sum_{i=1}^{n}y_{i}=\sum_{i=1}^{n}x_{i}$.