On a problem of T. Szostok concerning the Hermite-Hadamard inequalities

TL;DR

This paper characterizes solutions to a specific system of inequalities involving convex functions and their primitives, revealing that solutions are inherently regular without extra conditions.

Contribution

It provides a complete characterization of solutions to Szostok's inequality system, showing they must be continuous convex functions and their primitives.

Findings

Solutions are continuous convex functions.

F is a primitive of f.

Solutions exhibit inherent regularity.

Abstract

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.