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We generalize cyclic Jensen’s inequality utilizing the theory of 4-convex function under the effect of introduced Green functions. We formulate result for power means and quasi means. Also we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi-divergence, Shannon-entropy, Kullback-Leibler divergence and  χ  2 -divergence.

New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications