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 : → R is convex, then the following chain of inequalities hold:
The inequality has been generalized to higher dimensions: if is a bounded, convex domain and is a positive convex function, then
where is a constant depending only on the dimension.Suppose that, and choose distinct values from. Let be convex, and let denote the "integral starting at " operator; that is,
Then
Equality holds for all iff is linear, and for all iff is constant, in the sense that
The result follows from induction on.
