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)\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.

A corollary on Vandermonde-type integrals

Suppose that $-\infty < a < b < \infty$ , and choose $n$ distinct values ${x j} n j =1$ from $(a, b)$ . Let $f :[a, b] \to ℝ$ be convex, and let $I$ denote the "integral starting at $a$ " operator; that is,

(If)(x)=\int _{a}^{x}{f(t)\,dt}

.

Then

\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})

Equality holds for all ${x j} n j =1$ iff $f$ is linear, and for all $f$ iff ${x j} n j =1$ is constant, in the sense that

\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)!}}

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.