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[edit]

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[edit]

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.

v t e Convex analysis and variational analysis
Basic concepts	Convex combination Convex function Convex set
Topics (list)	Choquet theory Convex geometry Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex
Maps	Convex conjugate Concave (Closed K- Logarithmically Proper Pseudo- Quasi-) Convex function Invex function Legendre transformation Semi-continuity Subderivative
Main results (list)	Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality Hermite–Hadamard inequality Krein–Milman theorem Mazur's lemma Shapley–Folkman lemma Robinson–Ursescu Simons Ursescu
Sets	Convex hull (Orthogonally, Pseudo-) Convex set Effective domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope
Series	Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
Duality	Dual system Duality gap Strong duality Weak duality
Applications and related	Convexity in economics