In this article a new method is introduced to give weighted estimations of the differences of cyclic refinements of Jensen functionals. We utilize weighted form of Hermite–Hadamard (HH) inequality along with the approximations of Montgomery two- and one-point formulae with Peano type kernel along with the consequences of the n-times differentiable convex functions. As a result, we present new upper and lower bounds that are also verified with concrete examples to show the correctness of the bounds obtained for special cases. As a result, we provide estimations in terms of cyclic power means. We also improvise our results to give new uniform estimations for useful distance functions in information theory. Finally, some estimations of Zipf and Hybrid Zipf law are provided as well.