“Hermite–Hadamard inequality”的意思、由来-开放百科全书

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Hermite–Hadamard inequality

释义

Generalisations - The concept of a sequence of iterated integrals
Example 1 Example 2 Example 3 Theorem
References

{{Expert needed|Mathematics|talk=Generalisations - The concept of a sequence of iterated integrals|reason=The whole section was created by User Vezér who self-promoted in many ways his non-notable results in Wikipedia. I think this unnecessary section remained here from his work, and it is not really on the main topic of this article.|date=July 2018}}{{distinguish|Hadamard's 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:

Generalisations - The concept of a sequence of iterated integrals

Suppose that −∞ < a < b < ∞, and let f:[a, b] → ℝ be an integrable real function.

Under the above conditions the following sequence of functions is called the sequence of iterated integrals of f,where a ≤ s ≤ b.:

Example 1

Let [a, b] = [0, 1] and f(s) ≡ 1. Then the sequence of iterated integrals of 1 is defined on [0, 1], and

Example 2

Let [a,b] = [−1,1] and f(s) ≡ 1. Then the sequence of iterated integrals of 1 is defined on [−1, 1], and

Example 3

Let [a, b] = [0, 1] and f(s) = e^s. Then the sequence of iterated integrals of f is defined on [0, 1], and

Theorem

Suppose that −∞ < a < b < ∞, and let f:[a,b]→R be a convex function, a < x_i < b, i = 1, ..., n, such that x_i ≠ x_j, if i ≠ j. Then the following holds:

where

In the concave case ≤ is changed to ≥.

Remark 1. If f is convex in the strict sense then ≤ is changed to < and equality holds iff f is linear function.

Remark 2. The inequality is sharp in the following limit sense: let

and

Then the limit of the left side exists and

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}}

1 : Inequalities

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