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:

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

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

${\displaystyle {\frac {1}{|\Omega |}}\int _{\Omega }f(x)\,dx\leq {\frac {c_{n}}{|\partial \Omega |}}\int _{\partial \Omega }f(y)\,d\sigma (y)}$

where ${\displaystyle c_{n}}$ is a constant depending only on the dimension.

A corollary on Vandermonde-type integrals

This article needs attention from an expert in mathematics. The specific problem is: This section is not really on the main topic of this article.. See the talk page for details. WikiProject Mathematics may be able to help recruit an expert. (July 2018)

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

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

Then

${\displaystyle \sum _{i=1}^{n}{\frac {(I^{n-1}F)(x_{i})}{\prod _{j\neq i}{(x_{i}-x_{j})}}}\leq {\frac {1}{n!}}\sum _{i=1}^{n}f(x_{i})}$

Equality holds for all {x_j}ⁿ
_j=1 iff f is linear, and for all f iff {x_j}ⁿ
_j=1 is constant, in the sense that

${\displaystyle \lim _{\{x_{j}\}_{j}\to \alpha }{\sum _{i=1}^{n}{\frac {(I^{n-1}F)(x_{i})}{\prod _{j\neq i}{(x_{i}-x_{j})}}}}={\frac {f(\alpha )}{(n-1)!}}}$

The result follows from induction on n.

References

• Jacques Hadamard, "Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann", Journal de Mathématiques Pures et Appliquées, volume 58, 1893, pages 171–215.
• Zoltán Retkes, "An extension of the Hermite–Hadamard Inequality", Acta Sci. Math. (Szeged), 74 (2008), pages 95–106.
• Mihály Bessenyei, "The Hermite–Hadamard Inequality on Simplices", American Mathematical Monthly, volume 115, April 2008, pages 339–345.
• Flavia-Corina Mitroi, Eleutherius Symeonidis, "The converse of the Hermite-Hadamard inequality on simplices", Expo. Math. 30 (2012), pp. 389–396. doi: 10.1016/j.exmath.2012.08.011; ISSN  0723-0869
• Stefan Steinerberger, The Hermite-Hadamard Inequality in Higher Dimensions, The Journal of Geometric Analysis, 2019.