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. MSC: 26B25, 52A40.
The article provides refinements and generalizations of the Hermite-Hadamard inequality for convex functions on the bounded closed interval of real numbers. Improvements are related to the discrete and integral part of the inequality. MSC: 26A51, 26D15.
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.
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.
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...