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.
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