Hermite-Hadamard type inequalities by using Newton-Cotes quadrature formulas
Angshuman R. Goswami, Ferenc Hartung
TL;DR
The paper addresses estimating integral means for functions that are not strictly convex by introducing $\Phi$-monotone and $\Phi$-convex classes. It develops Hermite-Hadamard-type bounds for the numerical integral mean $\frac{1}{b-a}\mathcal{I}_{n}(f)$ based on Newton-Cotes quadrature formulas ($\mathcal{T}_{n}$, $\mathcal{S}_{n}$, and $\mathcal{S}_{n}^{\frac{3}{8}}$) and expresses bounds in terms of the error function $\Phi$ and endpoint values. Key contributions include parity-dependent corrections $E_n$ for the $\Phi$-convex case, refined bounds for $\Phi$-Hölder and $\Phi$-affine functions, and conditions (e.g., superadditivity of $\Phi$) under which $n$-dependence can be removed. In the smooth limit where $n^{2}\Phi(\frac{b-a}{n})\to0$, the classical Hermite-Hadamard inequality is recovered, providing a bridge between approximate convexity and numerical integration with practical implications for applications exhibiting near-convexity. The work offers quantitative tools for estimating integral means in settings such as finance or population dynamics where exact convexity fails.
Abstract
A convex function $f:[a,b]\to\mathbb{R}$ satisfies the so-called Hermite-Hadamard inequality $$ f\left(\frac{a+b}{2}\right)\leq \frac{1}{b-a}\int_a^{b}f(t)dt\leq \frac{f(a)+f(b)}{2}. $$ 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 $\frac1{b-a}\mathcal{I}_{n}(f)$, where $\mathcal{I}_{n}(f)$ denotes some of the most popular Newton-Cotes quadrature formulas.