New cyclic Jensen type bounds via weighted HH inequalities and Montgomery identity with application to information inequalities

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. •Provide bounds for cyclic jensen inequality via HH inequality and Montgomery identity.•We provide specific examples to show the correctness of obtained bounds.•We provide results for power means and quasi-arithmetic means.•We give applications in information theory.•Finally, we provide connections between Zipf and Hybrid Zipf-Mandelbrot entropies.