Majorization principle of Hardy-Littlewood-Pólya

The majorization principle of Hardy-Littlewood-Pólya ( English Hardy-Littlewood-Pólya majorization principle ) is a theorem of the mathematical branch of analysis , which emerges from a work of the three mathematicians Godfrey Harold Hardy , John Edensor Littlewood and George Pólya from 1929. It deals with conditions under which convex real functions satisfy a certain inequality . This inequality was in 1932, also from the Yugoslav mathematician Jovan Karamata found, they therefore also inequality of Karamata ( English inequality of Karamata is) called. Numerous mathematicians - such as László Fuchs and Alexander Markowitsch Ostrowski - have given generalizations, while Ky Fan and George G. Lorentz found a "continuous version" of them. The principle of majorization and the related results play an important role in matrix theory , probability theory and mathematical statistics .

formulation

The principle of majorization can be stated as follows:

Given a real interval and (for a natural number ) real numbers in it , so that the following inequalities are fulfilled: ${\displaystyle I}$ ${\displaystyle n\geq 2}$ ${\displaystyle x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}\in I}$

${\displaystyle {\begin{aligned}x_{1}&{\geq }\ldots \geq x_{n-1}\geq x_{n}\\y_{1}&{\geq }\ldots \geq y_{n-1}\geq y_{n}\\y_{1}&{\geq }x_{1}\\y_{1}+y_{2}&{\geq }x_{1}+x_{2}\\&{.}\\&{.}\\&{.}\\y_{1}+\dots +y_{n-1}&{\geq }x_{1}+\dots +x_{n-1}\\y_{1}+\dots +y_{n}&=x_{1}+\dots +x_{n}\\\end{aligned}}}$

Furthermore, let it be a continuous function whose restriction to the interior of the interval is Jensen-convex . ${\displaystyle f\colon I\to \mathbb {R} }$ ${\displaystyle f|_{I^{\circ }}}$

Then:

${\displaystyle {\begin{aligned}f(y_{1})+\dots +f(y_{n})&{\geq }f(x_{1})+\dots +f(x_{n})\\\end{aligned}}}$

Inference

With the principle of majorization, the following inequality can be obtained, which emerges from a work by VK Lim from 1971:

If above and if the real function fulfills the mentioned conditions, then the inequality always applies for every three real numbers ${\displaystyle I=[0,\infty )}$ ${\displaystyle f\colon I\to \mathbb {R} }$ ${\displaystyle a\geq 0,b\geq 0,c\geq a+b}$

${\displaystyle f(a)+f\left(b+c\right)\geq f\left(a+b\right)+f(c)}$  .

In the case of the function to the real exponents one speaks also of the inequality of Lim ( English Lim's inequality ). ${\displaystyle x\mapsto f(x)=x^{r}\;(x\geq 0)}$ ${\displaystyle r\geq 1}$

Sources and background literature

• 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 .
• Ky Fan, GG Lorentz: An integral inequality . In: American Mathematical Monthly . tape 61 , 1954, pp. 626-631 , doi : 10.2307 / 2307678 ( MR0064829 ).
• Ladislas Fuchs : A new proof of an inequality of Hardy-Littlewood-Pólya . In: Mat. Tidsskr. B . tape 1947 , 1947, pp. 53-54 ( MR0024480 ).
• GH Hardy, JE Littlewood, G. Pólya: Some simple inequalities satisfied by convex functions . In: The Messenger of Mathematics . tape 58 , 1929, pp. 145-152 .
• GH Hardy, JE Littlewood, G. Pólya: Inequalities . Reprint (of the 2nd edition 1952). Cambridge University Press , Cambridge 1964.
• J. Karamata: Sur une inégalité relative aux fonctions convexes . In: Publ. Math. Univ. Belgrade . tape 1 , 1932, p. 145-148 .
• 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 ).
• VK Lim: A note on an inequality . In: Nanta Mathematica . tape 5 , 1971, p. 38-40 ( MR0297949 ).
• 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 a. a. 1970, ISBN 3-540-62903-3 ( MR0274686 ).
• Alexandre Ostrowski : J. Math. Pures Appl. (9) . tape 31 , 1952, pp. 253-292 ( MR0052475 ).

Individual evidence

1. ^ Marek Kuczma: An Introduction to the Theory of Functional Equations and Inequalities. 2009, p. 211 ff.
2. ^ Edwin F. Beckenbach, Richard Bellman: Inequalities. 1983, p. 30 ff, p. 52 ff.
3. ^ GH Hardy, JE Littlewood, G. Pólya: Inequalities. 1964, p. 88 ff.
4. ↑ DS Mitrinović: Analytic Inequalities. 1970, p. 164 ff.
5. ↑ Kuczma, op.cit, p. 211.
6. ↑ Beckenbach, Bellman, op.cit, p. 30.
7. ↑ ^{a b} Kuczma, op.cit, p. 214.