Hermite–Hadamard inequality

Not to be confused with Hadamard's 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:

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

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

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

where $c_{n}$ is a constant depending only on the dimension. The best known bound on $c_{n}$ is $c_{n}\leq 2n^{3/2}$ and $c_{n}\geq n-1$ .

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.