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In the present paper we solve a problem posed by Tomasz Szostok who asked about the solutions $f$ and $F$ to the system of inequalities $$ f\Big(\frac{x+y}{2}\Big)\leq \frac{F(y)-F(x)}{y-x}\leq \frac{f(x)+f(y)}{2}. $$ We show that $f$ and $F$ are the solutions to the above system of inequalities if and only if $f$ is a continuous convex function and $F$ is primitive function of $f$. This result can be interpreted as a regularity phenomenon-the solutions to the system of functional inequalities turn out to be regular without any additional assumptions.

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.

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.