Wirtinger's inequality

The inequality of Wirtinger ( English Wirtinger's inequality ) is a classical inequalities of the mathematical area of analysis . It is named after the Austrian mathematician Wilhelm Wirtinger - apparently thanks to Wilhelm Blaschke's assignment in his monograph Circle and Ball from 1916 - although it is known that other mathematicians have previously presented similar inequalities that are valid under weaker conditions. Wirtinger's inequality gave rise to a large number of further investigations. Among other things, it is related to the Poincaré inequality .

formulation

The inequality can be stated as follows:

Let a real function with the following properties be given on the field of real numbers : ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$

(1) is a differentiable function . ${\displaystyle f}$

(2) is a periodic function of the period  . ${\displaystyle f}$ ${\displaystyle 2\pi }$

(3) The derivative function is a quadratic integrable function . ${\displaystyle f'\colon \mathbb {R} \to \mathbb {R} }$

(4) It is  . ${\displaystyle \int _{0}^{2\pi }{f(x)}\,dx=0}$

Then:

${\displaystyle \int _{0}^{2\pi }{[f(x)]^{2}\,dx}\leq \int _{0}^{2\pi }{[f'(x)]^{2}\,dx}}$  .

In this inequality, the equal sign applies if and only if there are real numbers such that the shape ${\displaystyle A,B}$ ${\displaystyle f}$

${\displaystyle f(x)=A\cos x+B\sin x\;(A,B\in \mathbb {R} )}$

Has.

Almansi's inequality

Even before 1916, the Italian mathematician Emilio Almansi found one of Wirtinger's closely related inequalities and published it in 1905:

Given are two real numbers and a real function with the following properties: ${\displaystyle a,b}$ ${\displaystyle f\colon \left[a,b\right]\to \mathbb {R} }$

(1) The restriction is continuously differentiable . ${\displaystyle f|_{(a,b)}}$

(2) It is . ${\displaystyle f(a)=f(b)}$

(3)  . ${\displaystyle \int _{a}^{b}{f(x)}\,dx=0}$

Then:

${\displaystyle \left({\frac {2\pi }{b-a}}\right)^{2}\cdot \int _{a}^{b}{[f(x)]^{2}\,dx}\leq \int _{a}^{b}{[f'(x)]^{2}\,dx}}$  .

Some authors, such as Leonida Tonelli in a paper from 1911, improved this inequality by Almansi by weakening the assumptions about the function . ${\displaystyle f}$

literature

• E. Almansi: Sopra una delle esperienze di Plateau . In: Annali di Matematica Pura ed Applicata, Series III . tape 12 , 1905, pp. 1-17 .
• Edwin F. Beckenbach , Richard Bellman : Inequalities (=  results of mathematics and their border areas . Volume 30 ). 4th edition. Springer Verlag , Berlin, Heidelberg, New York, Tokyo 1983, ISBN 3-540-03283-5 .
• DS Mitrinović : Analytic Inequalities . In cooperation with PM Vasić (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 165 ). Springer Verlag , Berlin ( inter alia ) 1970, ISBN 3-540-62903-3 ( MR0274686 ).
• L. Tonelli: Su una proposizione dell 'Almansi . In: Rendiconti della Regia Accademia dei Lincei . tape 23 , 1914, pp. 676-682 .
• Wilhelm Blaschke: circle and sphere . Veit, Leipzig 1916.

References and footnotes

1. ^ ^{A b} Edwin F. Beckenbach, Richard Bellman: Inequalities. 1983, pp. 177-185
2. ↑ ^{a b c} D. S. Mitrinović: Analytic Inequalities. 1970, pp. 141-154
3. ^ JB Diaz, FT Metcalf: Variations of Wirtinger's inequality. In: Oved Shisha (ed.): Inequalities: Proceedings of a Symposium Held at Wright-Patterson Air Force Base, Ohio, Aug. 19-27, 1965. Academic Press, New York, London (1967), pp. 73-77
4. ↑ See Wilhelm Blaschke: Circle and Ball. 1916, pp. 105-106: There Blaschke gives the inequality under the heading Ein Lemma von Wirtinger .