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 : [ab] → R is convex, then the following chain of inequalities hold:

$f\left({\frac {a+b}{2}}\right)\leq {\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx\leq {\frac {f(a)+f(b)}{2}}.$ 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.

## A corollary on Vandermonde-type integrals

Suppose that −∞ < a < b < ∞, and choose n distinct values {xj}n
j=1
from (a, b). Let f:[a, b] → be convex, and let I denote the "integral starting at a" operator; that is,

$(If)(x)=\int _{a}^{x}{f(t)\,dt}$ .

Then

$\sum _{i=1}^{n}{\frac {(I^{n-1}F)(x_{i})}{\prod _{j\neq i}{(x_{i}-x_{j})}}}\leq {\frac {1}{n!}}\sum _{i=1}^{n}f(x_{i})$ Equality holds for all {xj}n
j=1
iff f is linear, and for all f iff {xj}n
j=1
is constant, in the sense that

$\lim _{\{x_{j}\}_{j}\to \alpha }{\sum _{i=1}^{n}{\frac {(I^{n-1}F)(x_{i})}{\prod _{j\neq i}{(x_{i}-x_{j})}}}}={\frac {f(\alpha )}{(n-1)!}}$ The result follows from induction on n.