HLPK Type Theorem for Triangle Functionals Induced by Convex Functions and Further Results on Refining Jensen's Inequality

We study the triangle functionals induced by convex functions. To this end the vectorial majorization intended for comparing two tuples of vectors is used. A Hardy-Littlewood-Polya-Karamata (HLPK) type theorem is proved for the triangle functionals. As applications, some refinements of the celebrated Jensen's inequality are established. In doing so, the specification of HLPK Theorem for two triples of vectors is utilized. Special attention is paid to convex functions that are simultaneously concave (affine) on some subset of their domains. In particular, functions that are both convex and piecewise affine are considered. Some applications for convex sequences are also provided.