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:

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

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

$\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}f(x)dx\le \frac{c_n}{|\partial \mathrm{\Omega }|}{\int }_{\partial \mathrm{\Omega }}f(y)d\sigma (y)$

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

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. Template:Doi; Template:ISSN
• Stefan Steinerberger, The Hermite-Hadamard Inequality in Higher Dimensions, The Journal of Geometric Analysis, 2019.