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arXiv · 1808.06524

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

Andrzej Olbryś
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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.

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Andrzej Olbryś. 2018-08-16. On a problem of T. Szostok concerning the Hermite-Hadamard inequalities. https://arxiv.org/abs/1808.06524

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