Hermite–Hadamard inequality

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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 : [ab] → R is convex, 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[edit]

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

.

Then

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

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.