Refinements of the integral Jensen’s inequality generated by finite or infinite permutations

Abstract There are a lot of papers dealing with applications the so-called cyclic refinement discrete Jensen’s inequality. A significant generalization refinement, based on combinatorial considerations, has recently been discovered by author. In present paper we give integral versions these results. On one hand, new method to refine inequality is developed. other result contains some recent refinements as elementary cases. Finally Fejér (especially Hermite–Hadamard inequality), quasi-arithmetic means, and f -divergences presented.