Zlatko Pavić

The Inequalities for Quasiarithmetic Means

Overview and refinements of the results are given for discrete, integral, functional and operator variants of inequalities for quasiarithmetic means. The general results are applied to further refinements of the power means. Jensens inequalities have been systematically presented, from the older variants, to the most recent ones for the operators without operator convexity.

The Applications of Functional Variants of Jensen's Inequality

The paper is inspired by McShanes results on the functional form of Jensens inequality for convex functions of several variables. The work is focused on applications and generalizations of this important result. At that, the generalizations of Jensens inequality are obtained using the positive linear functionals.

Functions Like Convex Functions

The paper deals with convex sets, functions satisfying the global convexity property, and positive linear functionals. Jensens type inequalities can be obtained by using convex combinations with the common center. Following the idea of the common center, the functional forms of Jensens inequality are considered in this paper.

Convex combinations, barycenters and convex functions

The article first shows one alternative definition of convexity in the discrete case. The correlation between barycenters, Jensen’s inequality and convexity is studied in the integral case. The Hermite-Hadamard inequality is also obtained as a consequence of a concept of barycenters. Some derived results are applied to the quasi-arithmetic means and especially to the power means.MSC:26A51, 26D15, 28A10, 28A25.

Extension of Jensen’s inequality to affine combinations

The article provides the generalization of Jensen’s inequality for convex functions on the line segments. The main and preliminary inequalities are expressed in discrete form using affine combinations that can be reduced to convex combinations. The resulting quasi-arithmetic means are used to extend the two well-known inequalities.MSC:26A51, 26D15.

Inequalities for Convex Functions on Simplexes and Their Cones

The aim of this paper is to present the fundamental inequalities for convex functions on Euclidean spaces. The work is based on the geometry of the simplest convex sets and properties of convex functions. Some obtained inequalities are applied to demonstrate a natural way of generalizing the Hermite-Hadamard inequality.

Half convex functions

The paper provides the characteristic properties of half convex functions. The analytic and geometric image of half convex functions is presented using convex combinations and support lines. The results relating to convex combinations are applied to quasi-arithmetic means.MSC:26A51, 26D15.

Improvements of the Hermite-Hadamard inequality for the simplex

In this study, the simplex whose vertices are barycenters of the given simplex facets plays an essential role. The article provides an extension of the Hermite-Hadamard inequality from the simplex barycenter to any point of the inscribed simplex except its vertices. A two-sided refinement of the generalized inequality is obtained in completion of this work.

Application of Functionals in Creating Inequalities

The paper deals with the fundamental inequalities for convex functions in the bounded closed interval. The main inequality includes convex functions and positive linear functionals extending and refining the functional form of Jensen’s inequality. This inequality implies the Jensen, Fejer, and, thus, Hermite-Hadamard inequality, as well as their refinements.

Investigating an overdetermined system of linear equations by using convex functions

The paper studies the application of convex functions in order to prove the existence of optimal solutions of an overdetermined system of linear equations. The study approaches the problem by using even convex functions instead of projections. The research also relies on some special properties of unbounded convex sets, and the lower level sets of continuous functions.