Hermite–Hadamard inequality

In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ : [a, b] → R is convex, then the following chain of inequalities hold:


 * $$ f\left( \frac{a+b}{2}\right) \le \frac{1}{b - a}\int_a^b f(x)\,dx \le \frac{f(a) + f(b)}{2}. $$

The inequality has been generalized to higher dimensions: if $$ \Omega \subset \mathbb{R}^n $$ is a bounded, convex domain and $$f:\Omega \rightarrow \mathbb{R}$$ is a positive convex function, then


 * $$ \frac{1}{|\Omega|} \int_\Omega f(x) \, dx \leq \frac{c_n}{|\partial \Omega|} \int_{\partial \Omega} f(y) \, d\sigma(y) $$

where $$ c_n $$ is a constant depending only on the dimension.