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 f : [a, b] → R is convex, then the following chain of inequalities hold: f ( a + b 2 ) ≤ 1 b − a ∫ a b f ( x ) d x ≤ f ( a ) + f ( b ) 2 . {\displaystyle 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 Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} is a bounded, convex domain and f : Ω → R {\displaystyle f:\Omega \rightarrow \mathbb {R} } is a positive convex function, then 1 | Ω | ∫ Ω f ( x ) d x ≤ c n | ∂ Ω | ∫ ∂ Ω f ( y ) d σ ( y ) {\displaystyle {\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 {\displaystyle c_{n}} is a constant depending only on the dimension.

Source: Wikipedia — Hermite–Hadamard inequality (CC BY-SA 4.0)