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:

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

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

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

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

A corollary on Vandermonde-type integrals

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

$(If)(x)=\underset{a}{\overset{x}{\int }}f(t)dt$.

Then

$\sum _{i=1}^n\frac{(I^{n-1}F)(x_i)}{\prod _{j\ne i}(x_i-x_j)}\le \frac{1}{n!}\sum _{i=1}^{n}f(x_i)$

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

$\underset{\{x_j{\}}_j\to \alpha }{\lim }\sum _{i=1}^n\frac{(I^{n-1}F)(x_i)}{\prod _{j\ne i}(x_i-x_j)}=\frac{f(\alpha )}{(n-1)!}$

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.