On a problem of T. Szostok concerning the Hermite-Hadamard inequalities (Q2633723)

The author studies the system of functional inequalities \[ f\Big(\frac{x+y}{2}\Big)\le \frac{F(y)-F(x)}{y-x}\le \frac{f(x)+f(y)}{2},\quad x,y\in (a,b),\quad x\neq y. \] It is proved that $f$ and $F$ are the solutions of the above system of inequalities if and only if $f$ is a continuous convex function and $F$ is a primitive function of $f.$ This answers the question posed by \textit{E. Gselmann} (ed.) [Ann. Math. Sil. 28, 97--118 (2014; Zbl 1369.00112)] about the solutions $f$ and $F$ of the above system of inequalities. The 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.