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

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 ( x + y 2 ) ≤ F ( y ) − F ( x ) y − x ≤ 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.