New cyclic Jensen type estimations VIA weighted Hadamard inequalities under the effect of Fink's identity

Abstract

We take into account differences resulting from the discrete and integral cyclic Jensen's inequalities and provide upper and lower bounds using weighted Hermite-Hadamard inequalities with the support of Fink's identity. The theory of n-times differentiable convex functions is used to demonstrate this case. Our findings are applicable to any n ​≥ ​2, and we present specific examples to demonstrate the accuracy of the bounds found for unique situations. We also formulate results for power means and quasi-arithmetic means for higher order convexity. Last but not least, we provide applications in information theory by offering new uniform estimations of the generalized Csiszár, Rényi, Shannon, Kullback-Leibler, χ2 ​− ​divergence, Bhattacharyya, Zipf, and Hybrid Zipf-Mandelbrot entropies.