Search

Different terms such as variability, inequality and dispersion, which occur in various engineering problems and scientific fields, in mathematics are most simply described by the concept of majorization, a powerful mathematical tool which allows one to see the existing connections between vectors that can be used. In majorization theory, majorization inequalities play an important role. In this paper, using known properties of superquadratic functions, extensions and improvements of majorization inequalities are obtained. Also their converse inequalities are presented. For superquadratic functions, which are not convex, results analog ones for convex functions are presented. For superquadratic functions which are convex, improvements are given. At the end, applications to $$\varphi $$ -divergences are discussed. New estimates for the Renyi entropy are derivated.

An important class of Schur-convex functions is generated by convex functions via the well-known Hardy–Littlewood–Pólya–Karamata inequality. Sherman’s inequality is a natural generalization of the HLPK inequality. It can be viewed as a comparison of two special inner product expressions induced by a convex function of one variable. In the present note, we extend the Sherman inequality from the (bilinear) inner product to a (nonlinear) map of two vectorial variables satisfying the Leon–Proschan condition. Some applications are shown for directional derivatives and gradients of Schur-convex functions.

This paper gives extensions and improvements of Sherman’s inequality forn-convex functions obtained by using new identities which involve Green’s functions and Fink’s identity. Moreover, extensions and improvements of Majorization inequality as well as Jensen’s inequality are obtained as direct consequences. New inequalities between geometric, logarithmic and arithmetic means are also established.

Considering selfadjoint operators from C*-algebra on a Hilbert spaces and unital positive linear maps between C*-algebras we establish Sherman’s operator inequalities for operator convex functions. Using extended idea of convexity to operator functions of several variables we obtaine multidimensional Sherman’s operator inequality. Obtained result we apply to establish new results that include the well known operator means and relative operator entropy.

Considering the weighted concept of majorization, Sherman obtained generalization of majorization inequality for convex functions known as Sherman’s inequality. We extend Sherman’s result to the class of n-strongly convex functions using extended idea of convexity to the class of strongly convex functions. We also obtain upper bound for Sherman’s inequality, called the converse Sherman inequality, and as easy consequences we get Jensen’s as well as majorization inequality and their conversions for strongly convex functions. Obtained results are stronger versions for analogous results for convex functions. As applications, we introduced a generalized concept of f -divergence and derived some reverse relations for such concept.

In this paper we proved a converse to Sherman’s inequality. Using the concept of f-divergence we obtained some inequalities for the well-known entropies. We also introduced a new entropy by applying the Zipf-Mandelbrot law and derived some related inequalities

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.

New generalizations of Sherman's inequality for n-convex functions are obtained using Fink's identity and Green's function. Using inequalities for the Čebyšev functional, some new Ostrowsky and Grüss type inequalities related to these generalizations are established.