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:
- [math]\displaystyle{ 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}. }[/math]
The inequality has been generalized to higher dimensions: if [math]\displaystyle{ \Omega \subset \mathbb{R}^n }[/math] is a bounded, convex domain and [math]\displaystyle{ f:\Omega \rightarrow \mathbb{R} }[/math] is a positive convex function, then
- [math]\displaystyle{ \frac{1}{|\Omega|} \int_\Omega f(x) \, dx \leq \frac{c_n}{|\partial \Omega|} \int_{\partial \Omega} f(y) \, d\sigma(y) }[/math]
where [math]\displaystyle{ c_n }[/math] is a constant depending only on the dimension.
A corollary on Vandermonde-type integrals
Suppose that −∞ < a < b < ∞, and choose n distinct values {xj}nj=1 from (a, b). Let f:[a, b] → ℝ be convex, and let I denote the "integral starting at a" operator; that is,
- [math]\displaystyle{ (If)(x)=\int_a^x{f(t)\,dt} }[/math].
Then
- [math]\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) }[/math]
Equality holds for all {xj}nj=1 iff f is linear, and for all f iff {xj}nj=1 is constant, in the sense that
- [math]\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)!} }[/math]
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.
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