Some counterpart inequalities for a functional associated with Jensen's inequality
The authors offer properties of \[ \sum_{i\in I} p_if({\mathbf x}_i)-P_If(\sum_{i\in I}p_i{\mathbf x}_i/P_I). \] Here \(f\) is supposed to be a convex (and most of the time differentiable) real valued function on a convex subset \(C\) of a (mostly finite dimensional) linear space, \({\mathbf x}_i\in C\), the \(p_i\) are positive reals, \(i\in I\), \(I\) is a finite set of positive integers and \(P_I=\sum_{i\in I} p_i\) (though the last equation seems to have been omitted from the paper or thought to go without saying). Applications to arithmetic and geometric means and to the Shannon inequality are offered.
scientific article; zbMATH DE number 1086031