The Hermite–Hadamard inequality in Beckenbach's setting

Abstract

The classical Hermite-Hadamard inequality gives a lower and an upper estimations for the integral average of convex functions defined on compact intervals, involving the midpoint and the endpoints of the domain. The aim of the present paper is to extend this inequality and to give analogous results when the convexity notion is induced by Beckenbach families.The key tool of the investigations is based on some general support theorems that are obtained via the pure geometric properties of Beckenbach families and can be considered as generalizations of classical support and chord properties of ordinary convex functions. The Markov-Krein-type representation of Beckenbach families is also investigated.

Fractional Integral Inequalities for Differentiable Convex Mappings and Applications to Special Means and a Midpoint Formula

“…For more recent results which generalize, improve, and extend the inequalities presented above, one can see Abramovich et al [10], Cal et al [11], Avci et al [12], Odemir et al [13; 14], Dragomir [15; 16], Sarikaya et al [17], Xiao et al [18], Xi et al [19], Bessenyei [20], Tseng et al [21], Niculescu [22] and references therein.…”

Support-type properties of generalized convex functions

“…Let us turn attention to support-type results of the paper [4] by Bessenyei. He obtained the existence of principal supports in the setting of convexity with respect to so-called Beckenbach families, which includes U -convexity as a special case (this is a kind of convexity considered by Tornheim [22]-cf.…”

“…For instance they can be used to prove Hermite-Hadamard-type inequalities between quadrature operators (the well known Gauss-Legendre, Lobatto and Radau quadratures in the case of higher-order convexity) and the integral of the U -convex function. For details see [24] (higherorder convexity) and [4] (the general U -convexity).…”

“…, 2, 1) for even n (with the reservation that the points of multiplicity 1 are the endpoints of I ). Such supports, following Bessenyei [4], will be called principal supports. Proof.…”

On some fractional Hermite–Hadamard inequalities via s-convex and s-Godunova–Levin functions and their applications

“…For recent results concerning fractional Hermite-Hadamard inequalities and concepts of all kinds of convexity see [1][2][3]6,8,10,13,15,16,18] and references therein. Very recently, Tunç [17] established the following equality.…”