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

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.