On Jensen's inequality for a class of functions of Godunova and Levin
The authors investigate inequalities for functions \(f:C\to\hbox{\mathbf R}\) satisfying \[ f(\lambda x+(1-\lambda)y)\leq {f(x)\over \lambda}+{f(y)\over1-\lambda} \] for all \(x,y\in C\) and \(\lambda\in(0,1)\) (where \(C\) denotes a convex set of a real linear space). These functions are said to belong to the class \(Q\) by the terminology of \textit{E.K. Godunova} and \textit{V.I. Levin} [``Inequalities for functions of a broad class that contains convex monotone and some other forms of functions'' (Russian), Numer. Math. and Math. Phys. 138-142, 166 (1985; MR 88h:26013)]. The main results of the paper offer inequalities for the comparison of expressions of the following form \[ \sum_{i_1,\dots,i_k=1}^n {f(q_1x_{i_1}+\cdots+q_kx_{i_k}) \over p_{i_1}\cdots p_{i_k}}, \] where \(\bar{p}=(p_1,\dots,p_n)\) is a positive probability distribution, \(\bar{q}=(q_1,\dots,q_k)\) is a nonnegative probability distribution and \(x_1,\dots,x_n\in C\).
scientific article; zbMATH DE number 1101397