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:

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

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

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

where ${\displaystyle c_{n}}$ 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}ⁿ
_j=1 from (a, b). Let f:[a, b] → ℝ be convex, and let I denote the "integral starting at a" operator; that is,

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

Then

${\displaystyle \sum _{i=1}^{n}{\frac {(I^{n-1}F)(x_{i})}{\prod _{j\neq i}{(x_{i}-x_{j})}}}\leq {\frac {1}{n!}}\sum _{i=1}^{n}f(x_{i})}$

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

${\displaystyle \lim _{\{x_{j}\}_{j}\to \alpha }{\sum _{i=1}^{n}{\frac {(I^{n-1}F)(x_{i})}{\prod _{j\neq i}{(x_{i}-x_{j})}}}}={\frac {f(\alpha )}{(n-1)!}}}$

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.