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''nbsp;→ R is convex, then the following chain of inequalities hold:

: $f\left( \frac\right) \le \frac\int_a^b f(x)\,dx \le \frac.$

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

: $\frac \int_\Omega f(x) \, dx \leq \frac \int_ f(y) \, d\sigma(y)$

where $c_n$ is a constant depending only on the dimension.

A corollary on Vandermonde-type integrals

Suppose that , and choose distinct values from . Let be convex, and let denote the "integral starting at " operator; that is,
:$(If)(x)=\int_a^x$.

Then

: $\sum_^n \frac \leq \frac \sum_^n f(x_i)$

Equality holds for all iff is linear, and for all iff is constant, in the sense that
: $\lim_=\frac$

The result follows from induction on .

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. ;
* Stefan Steinerberger, The Hermite-Hadamard Inequality in Higher Dimensions, The Journal of Geometric Analysis, 2019.

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Inequalities
Theorems involving convexity