The Hermite-Hadamard inequality on hypercuboid

Abstract

Given any $\mathbf{a}:=\left(a_1,a_2,\dots ,a_n\right)$ and $\mathbf{b}:=\left(b_1,b_2,\dots ,b_n\right)$ in ${\mathbb{R}}^n$. The $\text{n}$-fold convex function defined on $\left[\mathbf{a},\mathbf{b}\right]$, $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^n$ with $\mathbf{a}<\mathbf{b}$ is a convex function in each variable separately. In this work we prove an inequality of Hermite-Hadamard type for $\text{n}$-fold convex functions. Namely, we establish the inequality \begin{align*} f\left( {\frac{{\bf{a} + {\bf{b}}}}{2}} \right) \le \frac{1}{{\bf{b} - {\bf{a}}}}\int_{\bf{a}}^{\bf{b}} {f\left( {\bf{x}} \right)d{\bf{x}}} \le \frac{1}{2^n }\sum\limits_{\bf{c}} {f\left( {\bf{c}} \right)}, \end{align*} where $\sum _{\mathbf{c}}f\left(\mathbf{c}\right):=\sum _{\underset{1\le i\le n}{c_i\in \left\{a_i,b_i\right\}}}f\left(c_1,c_2,\dots ,c_n\right)$. Some other related result are given.

12 pages