Hadamard's integral inequality

The Integralungleichung of Hadamard ( English Hadamard's integral inequality ) or inequality of Hadamard ( English Hadamard inequality ) is one of the classic inequalities of mathematics and as such the Analysis belong. It goes back to a publication by the French mathematician Jacques Salomon Hadamard from 1893 and gives an upper and lower estimate for the integrals of certain convex functions . Hadamard's integral inequality gave rise to numerous investigations and generalizations.

formulation

The inequality can be stated as follows:

Given a compact interval in the field of real numbers and then a continuous function whose restriction to the interior of the interval should also be Jensen-convex . ${\displaystyle J=[a,b]\;(a,b\in \mathbb {R} \;,a<b)}$ ${\displaystyle f\colon J\to \mathbb {R} }$ ${\displaystyle f|_{J^{\circ }}}$

Then:

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

Remarks

• Some authors denote the Hadamard Integralungleichung - with additional reference to the Danish mathematician Johan Jensen - as inequality of Jensen-Hadamard ( English Jensen-Hadamard inequality ). It should be noted that the anterior estimate of the integral inequality results from a simple application of the continuous variant of Jensen's inequality .
• If the Integralungleichung often also called Inequality of Hermite-Hadamard ( English Hermite-Hadamard inequality ) as they essentially - in fact even in 1881! - was found (and even announced) by the French mathematician Charles Hermite . However, this was initially ignored, as was the publication presented by Hermite in Mathesis in 1883 .
• The integral inequality can be seen as the starting point of Choquet's theory . Within the framework of this theory it can be shown that under the conditions described in Choquet's theorem , an analogous integral inequality holds. In particular, this analog is valid for every - dimensional simplex and every borelian probability measure defined on . ${\displaystyle n}$ ${\displaystyle \Delta \subset \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle \Delta }$

application

By applying the integral inequality to the real function , the following inequality can be derived, as Hermite showed in his work from 1883: ${\displaystyle x\mapsto f(x)={\frac {1}{1+x}}\;(x\geq 0)}$

${\displaystyle x-{\frac {x^{2}}{2+x}}<\ln(1+x)<x-{\frac {x^{2}}{2+2x}}}$  .

This results for all natural numbers ${\displaystyle n>0}$

${\displaystyle {\frac {1}{n+{\tfrac {1}{2}}}}<\ln(n+1)-\ln n<{\frac {1}{2}}\left({\frac {1}{n}}+{\frac {1}{n+1}}\right)}$  .

The latter inequality leads to a derivation of the Stirling formula .

Sources and background literature

• SS Dragomir: Further properties of some mappings associated with Hermite-Hadamard inequalities . In: Tamkang Journal of Mathematics . tape 34 , 2003, p. 45-57 ( MR1960410 ).
• Ivan B. Lacković, Miomir S. Stanković: On Hadamard's integral inequality for convex functions . In: Univ. Beograd. Publ. Elektrotehn. Fac. Ser. Mat. Fiz. No. 412-460 , 1973, pp. 89-92 ( MR0349934 ).
• J. Hadamard: Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann . In: J. Math. Pures Appl. (4 ^e série) . tape 9 , 1893, pp. 171-215 .
• Ch. Hermite: Sur deux limites d'une intégrale définie . In: Mathesis . tape 3 , 1883, p. 82 .
• Marek Kuczma : An Introduction to the Theory of Functional Equations and Inequalities . Cauchy's Equation and Jensen's Inequality. 2nd Edition. Birkhäuser Verlag , Basel 2009, ISBN 978-3-7643-8748-8 ( MR2467621 ).
• Constantin Niculescu , Lars-Erik Persson : Convex Functions and Their Applications . A Contemporary Approach (=  CMS Books in Mathematics . Volume 23 ). Springer Verlag , New York 2006, ISBN 0-387-24300-3 ( MR2178902 ).
• Constantin P. Niculescu: The Hermite-Hadamard inequality for convex functions of a vector variable . In: Mathematical Inequalities & Applications . tape 5 , 2002, p. 619-623 ( MR1931222 ).
• Zoltán Retkes: An extension of the Hermite-Hadamard inequality . In: Acta Sci. Math. (Szeged) . tape 74 , 2008, p. 95-106 ( MR2431093 ).

Individual evidence

1. ^ Marek Kuczma: An Introduction to the Theory of Functional Equations and Inequalities. 2009, p. 215 ff
2. ↑ Kuczma, op. Cit. , P. 216
3. ^ Constantin P. Niculescu, Lars-Erik Persson: Convex Functions and Their Applications. 2006, p. 50 ff
4. ↑ Niculescu / Persson, op.cit., P. 52, p. 177 ff
5. ↑ Niculescu / Persson, op.cit., Pp. 193 ff
6. ^ Constantin P. Niculescu: The Hermite-Hadamard inequality for convex functions of a vector variable. Math. Inequal. Appl. 5 (2002), pp. 619-623
7. ↑ ^{a b} Niculescu / Persson, op.cit., P. 51