Schweitzer's inequality

The inequality of Schweitzer ( English Schweitzer’s inequality ) is an inequality of the mathematical field of analysis and in a certain way complementary to the inequality of Cauchy-Schwarz . She goes to a work of Pál Schweitzer from 1914 back, as a result, a number of further studies, joined to those who further inequalities provided the same type. Closely related to this inequality is not least the Kantorovich inequality presented by the Soviet mathematician Leonid Witaljewitsch Kantorowitsch in 1948 . With the Schweitzer inequality one obtains, among other things, certain upper estimates for the arithmetic mean values of finitely many positive numbers .

formulation

The inequality says the following:

A real interval to two positive numbers and further a natural number and positive numbers are given . ${\displaystyle [m,M]\subset \mathbb {R} ^{>0}}$ ${\displaystyle m\leq M}$ ${\displaystyle N>0}$ ${\displaystyle N}$ ${\displaystyle a_{1},\ldots ,a_{N}\in [m,M]}$

Then:

${\displaystyle \left({\frac {1}{N}}\sum _{k=1}^{N}{a_{k}}\right)\cdot \left({\frac {1}{N}}\sum _{k=1}^{N}{\frac {1}{a_{k}}}\right)\leq {\frac {(m+M)^{2}}{4mM}}}$  .

If any real interval and a real function are given and if the associated real function can be integrated , then the integral inequality applies ${\displaystyle [a,b]\subseteq \mathbb {R} }$ ${\displaystyle f\colon [a,b]\to [m,M]}$ ${\displaystyle f}$ ${\displaystyle x\mapsto {\frac {1}{f(x)}}}$

${\displaystyle \left({\frac {1}{b-a}}\int _{a}^{b}{f(x)}\,dx\right)\cdot \left({\frac {1}{b-a}}\int _{a}^{b}{\frac {1}{f(x)}}\,dx\right)\leq {\frac {(m+M)^{2}}{4mM}}}$  .

generalization

In Volume I of their two-volume textbook, Tasks and Theorems from Analysis , Georg Pólya and Gábor Szegö present an extensive generalization of the Schweitzer inequality:

Given are two real intervals with four positive numbers and a natural number as well as positive numbers and . ${\displaystyle [m_{1},M_{1}],[m_{2},M_{2}]\subset \mathbb {R} ^{>0}}$ ${\displaystyle m_{1}\leq M_{1},m_{2}\leq M_{2}}$ ${\displaystyle N>0}$ ${\displaystyle 2N}$ ${\displaystyle a_{1},\ldots ,a_{N}\in [m_{1},M_{1}]}$ ${\displaystyle b_{1},\ldots ,b_{N}\in [m_{2},M_{2}]}$

Then:

${\displaystyle 1\leq {\frac {\left(\sum _{k=1}^{N}{{a_{k}}^{2}}\right)\cdot \left(\sum _{k=1}^{N}{{b_{k}}^{2}}\right)}{\left(\sum _{k=1}^{N}{a_{k}b_{k}}\right)^{2}}}\leq \left({\frac {{\sqrt {\frac {M_{1}M_{2}}{m_{1}m_{2}}}}+{\sqrt {\frac {m_{1}m_{2}}{M_{1}M_{2}}}}}{2}}\right)^{2}}$  .

If an arbitrary real interval and two integrable real functions and are given, then is ${\displaystyle [a,b]\subseteq \mathbb {R} }$ ${\displaystyle f_{1}\colon [a,b]\to [m_{1},M_{1}]}$ ${\displaystyle f_{2}\colon [a,b]\to [m_{2},M_{2}]}$

${\displaystyle 1\leq {\frac {\left(\int _{a}^{b}{[f_{1}(x)]^{2}\,dx}\right)\cdot \left(\int _{a}^{b}{[f_{2}(x)]^{2}\,dx}\right)}{\left(\int _{a}^{b}{f_{1}(x)f_{2}(x)\,dx}\right)^{2}}}\leq \left({\frac {{\sqrt {\frac {M_{1}M_{2}}{m_{1}m_{2}}}}+{\sqrt {\frac {m_{1}m_{2}}{M_{1}M_{2}}}}}{2}}\right)^{2}}$  .

annotation

Some authors refer to the inequality of Schweitzer, the above-mentioned inequality polyA-Szegö and also inequalities similar type as the Cauchy-Schwarz inequality complementary inequalities ( English complementary inequalities ).

literature

• JB Diaz, FT Metcalf: Inequalities complementary to Cauchy's inequality for sums of real numbers . 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, p. 73-77 ( MR0222228 ).
• 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 ).
• Georg Pólya, Gábor Szegö: Exercises and theorems from analysis . Volume I: Series, Integral Calculus, Function Theory (=  Heidelberger Taschenbücher . Volume 73 ). 4th edition. Springer Verlag , Berlin 1970 ( MR0271277 ).
• P. Schweitzer: Egy egyenlötlenség az aritmetikai középértékröl [ An inequality related to the arithmetic mean ] . In: Math. És phys. Lapok . tape 23 , 1914, pp. 257-261 .
• 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.

Web links

• Schweitzer Inequality and Evidence on cut-the-knot.org

Individual evidence

1. ↑ See discussion! It may be the mechanical engineer Paul Henry Schweitzer (1893–1980) , who came from Hungary and later emigrated to the USA .
2. ↑ DS Mitrinović: Analytic Inequalities. 1970, pp. 59-66
3. ^ Georg Pólya, Gábor Szegö: Exercises and theorems from Analysis, Vol. I. 1970, p. 57, pp. 213-214
4. ↑ ^{a b} J. B. Diaz, FT Metcalf: Inequalities complementary to Cauchy's inequality for sums of real numbers. 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
5. ↑ Mitrinović, op.cit., P. 59
6. ↑ Pólya / Szegö, op. Cit. P. 57
7. ↑ Mitrinović, op.cit., P. 60
8. ↑ Diaz / Metcalf, op.cit., P. 74
9. ↑ Here and before, the front inequality corresponds to the inequality of Cauchy-Schwarz.