In majorization theory, the well-known majorization theorem plays a very important role. A more general result was obtained by Sherman. In this paper, concerning 2n-convex functions, we get generalizations of these results applying Lidstone's interpolating polynomials and theČebyšev functional. Using the obtained results, we generate a new family of exponentially convex functions. The results are some new classes of two-parameter Cauchy type means.
MSC: 26D15
In this paper we give extensions of Sherman’s inequality considering the class of convex functions of higher order. As particular cases, we get an extended weighted majorization inequality as well as Jensen’s inequality which have direct connection to information theory. We use the obtained results to derive new estimates for Shannon’s and Rényi’s entropy, information energy, and some well-known measures between probability distributions. Using the Zipf–Mandelbrot law, we introduce new functionals to derive some related results.
Considering a weighted relation of majorization, Sherman obtained a useful generalization of the classical majorization inequality. The aim of this paper is to extend Sherman's inequality to convex functions of higher order. An upper bound for Sherman's inequality, as well as for generalized Sherman's inequality, is given with some applications. As easy consequences, some new bounds for Jensen's inequality can be derived.
MSC: 26D15; 26D20; 26D99
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