\(s\)-Orlicz convex functions in linear spaces and Jensen's discrete inequality
The authors generalize apparently unpublished results of H. Hudzik and L. Maligranda (Preprint 02-1992, LuleåUniv. of Technology, Sweden) from the reals to linear spaces. They call a subset \(K\) of a linear space ``\(s\)-Orlicz-convex'' if, with every \(x\) and \(y\) in \(K\) also \(\alpha x+\beta y\in K\) for all nonnegative \(\alpha,\beta\) satisfying \(\alpha^{s}+\beta^{s} = 1\). A real valued mapping on \(K\) is \(s\)-Orlicz-convex if for all such \(x,y,\alpha,\beta\) we have \(f(\alpha x + \beta y) \leq \alpha^{s}f(x) + \beta^{s}f(y)\). The authors offer, among others, several generalizations of this Jensen inequality.
scientific article; zbMATH DE number 1046937