In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hermite–Hadamard inequality, named after
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
and
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...
and sometimes also called Hadamard's inequality, states that if a function ƒ :
'a'', ''b''nbsp;→ R is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
, then the following chain of inequalities hold:
:
The inequality has been generalized to higher dimensions: if
is a bounded, convex domain and
is a positive convex function, then
:
where
is a constant depending only on the dimension.
A corollary on Vandermonde-type integrals
Suppose that , and choose distinct values from . Let be convex, and let denote the
"integral starting at " operator; that is,
:
.
Then
:
Equality holds for all iff is linear, and for all iff is constant, in the sense that
:
The result follows from induction on .
References
*
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...
, "Étude sur les propriétés des
fonctions entières et en particulier d'une fonction considérée par
Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
", ''
Journal de Mathématiques Pures et Appliquées
The ''Journal de Mathématiques Pures et Appliquées'' () is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874). The journal was originally published by Charles Louis Étienne Bachelier. A ...
'', volume 58, 1893, pages 171–215.
* Zoltán Retkes, "An extension of the Hermite–Hadamard
Inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
", ''
Acta Sci. Math. (Szeged)'', 74 (2008), pages 95–106.
* Mihály Bessenyei, "The Hermite–Hadamard
Inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
on
Simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
", ''
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.
The ''American Mathematical Monthly'' is an e ...
'', 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. ;
* Stefan Steinerberger, The Hermite-Hadamard Inequality in Higher Dimensions, The Journal of Geometric Analysis, 2019.
{{DEFAULTSORT:Hermite-Hadamard inequality
Inequalities
Theorems involving convexity