On Jessen's inequality for convex functions (Q1074731)

Let E be a nonempty set and let L be a linear space of real-valued functions on E; suppose that the constant function 1 belongs to L. Let \(\phi\) be a convex function on an interval \(I\subset R\) and let A be a positive linear functional on L, with \(A(1)=1.\) Then for all \(g\in L\) such that \(\phi (g)\in L\) we have \(A(g)\in I\) and \(\phi (A(g))\leq A(\phi (g))\) (Jessen's inequality). The authors give a short proof of this inequality and prove some general complementary inequalities. They obtain Hölder's and Minkowski's inequality for positive linear functionals, as well as certain complementary Hölder and Minkowski inequalities. Other known results are generalized and some examples and applications are given.