Schur's inequality

The inequality of Schur ( English Schur’s inequality ) is one of several classic inequalities that the mathematician Issai Schur contributed to the mathematical field of analysis .

Representation of the inequality

The inequality is as follows:

Let real numbers be given and hold . ${\displaystyle r,x,y,z}$ ${\displaystyle x,y,z>0}$

Then there is the inequality

${\displaystyle x^{r}(x-y)(x-z)+y^{r}(y-z)(y-x)+z^{r}(z-x)(z-y)\geq 0}$

and the equals sign applies if and only if the three numbers all match. ${\displaystyle x,y,z}$

application

Using the Schurian inequality above , one of the numerous geometric inequalities in the triangular geometry of the Euclidean plane can be derived: ${\displaystyle r=2}$

If any triangle is given in the Euclidean plane , the sides of which should have the lengths , and is here equal to half the circumference of , then the inequality always applies ${\displaystyle ABC}$ ${\displaystyle a,b,c}$ ${\displaystyle s}$ ${\displaystyle ABC}$

${\displaystyle abcs\geq a^{3}\cdot (s-a)+b^{3}\cdot (s-b)+c^{3}\cdot (s-c)}$

literature

• Claudi Alsina , Roger B. Nelsen : When Less is More: Visualizing Basic Inequalities (=  The Dolciani Mathematical Expositions . Volume 36 ). The Mathematical Association of America , Washington, DC 2009, ISBN 978-0-88385-342-9 ( MR2498836 ).
• GH Hardy , JE Littlewood , G. Pólya : Inequalities . Reprint (of the 2nd edition 1952). Cambridge University Press , Cambridge 1973.
• 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 ).

Individual evidence

1. ↑ ^{a b} D. S. Mitrinović: Analytic Inequalities. 1970, p. 119 ff
2. ^ ^{A b} G. H. Hardy, JE Littlewood, G. Pólya: Inequalities. 1964, p. 64
3. ^ ^{A b} Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. 2009, pp. 37-38
4. ↑ Alsina / Nelsen, op.cit., P. 38