Support theorems in abstract settings

In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain space, we allow algebraic structures equipped with families of algebraic operations whose operations are mutually distributive with respect to each other. We introduce several new concepts in such algebraic structures, the notions of convex set, extreme set, and interior point with respect to a given family of operations, furthermore, we describe their most basic and required properties. In the context of the range space, we introduce the notion of completeness of a partially ordered set with respect to the existence of the infimum of lower bounded chains, we also offer several sufficient condition which imply this property. For instance, the order generated by a sharp cone in a vector space turns out to possess this completeness property. By taking several particular cases, we deduce support and extension theorems in various classical and important settings

On the K-Riemann integral and Hermite–Hadamard inequalities for K-convex functions

In the present paper we introduce a notion of the K-Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the K-Riemann integral and the convexity notion is replaced by K-convexity

On sandwich theorem for delta-subadditive and delta-superadditive mappings

In the present paper, inspired by methods contained in Gajda and Kominek (Stud Math 100:25–38, 1991) we generalize the well known sandwich theorem for subadditive and superadditive functionals to the case of delta-subadditive and delta-superadditive mappings. As a consequence we obtain the classical Hyers–Ulam stability result for the Cauchy functional equation. We also consider the problem of supporting delta-subadditive maps by additive ones

Inequalities of the Hermite-Hadamard type involving numerical differentiation formulas

We observe that the Hermite-Hadamard inequality written in the form $$f\left(\frac{x+y}{2}\right)\leq\frac{F(y)-F(x)}{y-x}\leq\frac{f(x)+f(y)}{2}$$ may be viewed as an inequality between two quadrature operators $f\left(\frac{x+y}{2}\right)$ $\frac{f(x)+f(y)}{2}$ and a differentiation formula $\frac{F(y)-F(x)}{y-x}.$ We extend this inequality, replacing the middle term by more complicated ones. As it turns out in some cases it suffices to use Ohlin lemma as it was done in a recent paper \cite{Rajba} however to get more interesting result some more general tool must be used. To this end we use Levin-Ste\v{c}kin theorem which provides necessary and sufficient conditions under which inequalities of the type we consider are satisfied

On a separation theorem for delta-convex functions

In the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given

Systems of Inequalities Characterizing Ring Homomorphisms

Assume that T : P -> R and U : P -> R are arbitrary mappings between two partially ordered rings P and R. We study a few systems of functional inequalities which characterize ring homomorphisms. For example, we prove that if T and U satisfy T(f+g) >= T(f)+T(g), U(f.g) >= U(f).U(g), for all f,g is an element of P and T >= U, then U = T and thismapping is a ring homomorphism. Moreover, we find two other systems for which we obtain analogous assertions

Characterization of t-affine differences and related forms

In the present paper we are concerned with the problem of characterization of maps which can be expressed as an affine difference i.e. a map of the form tf(x) + (1 − t)f(y) − f(tx + (1 − t)y), where t ∈ (0, 1) is a given number. We give a general solution of the functional equation associated with this problem

On T -Schur Convex Maps

We introduce and examine the notion of T-Schur convexity which is naturally connected with Schur convexity. As a particular case, we consider T-Wright convex maps which generalize a well-known and intensively investigated class of t-Wright convex functions. We discuss several properties of this class of functions. In the last part of the paper we give a characterization of T-Wright affine maps i.e. maps satisfying the corresponding functional equation

A characterization of (t1,…,tn)(t_1,\dots,t_n)-Wright affine functions

In 1998 K.Lajkó [5] gave a characterization of $t$−Wright affine functions. Now, we extend this result to $(t_1 ,\dots, t_n)$-Wright affine functions of an arbitrary order