The idea of the well known Jensen inequality is to interpolate convex functions, by finding an upper bound of the function at a point in the convex hull of predefined points. In this article, we present a counterpart of this inequality by giving a lower bound of the function outside this convex hull. This inequality is then refined by finding as many refining positive terms as we wish. Some applications treating means, integrals and eigenvalues are given in the end. Moreover, we present a MATLAB code that helps generate the parameters appearing in our results.