Hermite-Hadamard type inequalities by using Newton-Cotes quadrature formulas

A convex function $f:[a,b]\to \mathbb{R}$ satisfies the so-called Hermite-Hadamard inequality $$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_a^{b}f(t)dt\le \frac{f(a)+f(b)}{2}\mathrm{.}$$ Motivated by the above estimates, in this paper we consider approximately monotone and convex functions, and give upper and lower bounds to the numerical integral mean, i.e., to $\frac{1}{b-a}{\mathcal{I}}_n(f)$, where ${\mathcal{I}}_n(f)$ denotes some of the most popular Newton-Cotes quadrature formulas.