On a general inequality with applications (Q686868)

The following is a typical result offered. If \(f\) and \(g\) are real- resp. positive valued additive functions (misleadingly called affine in the paper) on an abelian group and \(F\) is real-valued and convex on the codomain of \(f\) then \[ g(x+y)F(f(x+y)/g(x+y))\leq g(x)F(f(x)/g(x))+g(y)F(f(y)/g(y)). \] Examples and generalizations to set- valued functions follow.