Inequality of Guha

The inequality of Guha ( English Guha’s inequality ) is one of several elementary inequalities in the area of the AGM inequality and as such can be assigned to the mathematical field of analysis . It goes back to a scientific publication by UC Guha from 1967.

Representation of the inequality

The inequality is as follows:

Let real numbers be given and for these hold as well as and . ${\ displaystyle a, p, q, x, y}$ ${\ displaystyle a \ geq 0}$ ${\ displaystyle p \ geq q> 0}$ ${\ displaystyle x \ geq y> 0}$

Then:

{\ displaystyle {\ bigl (} px + y + a {\ bigr)} \ cdot {\ bigl (} x + qy + a {\ bigr)} \ geq {\ bigl (} (p + 1) x + a {\ bigr)} \ cdot {\ bigl (} (q + 1) y + a {\ bigr)}}

Remarks

The importance of the inequality is that, as Guha showed in 1967, it enables a simple and at the same time clever derivation of the AGM inequality for any (but finite ) number of nonnegative numbers .
The proof of the inequality can be done purely algebraically . By means of algebraic transformations one can prove their equivalence with the inequality , which is obviously valid due to the assumptions made. It can also be shown in a geometrically illustrative manner. ${\ displaystyle \ left (px-qy \ right) \ cdot \ left (xy \ right) \ geq 0}$

The equal sign applies exactly in the case .

{\ displaystyle x = y}

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 ).
PS Bullen , DS Mitrinović , Petar M. Vasić (Eds.): Means and Their Inequalities (= Mathematics and Its Applications (East European Series) . Volume 31. ). D. Reidel Publishing Company , Dordrecht 1988, ISBN 90-277-2629-9 ( MR0947142 - Translated and revised from the Serbo-Croatian).
UC Guha: Arithmetic mean - geometric mean inequality . In: Mathematical Gazette . tape 51 , 1967, p. 145-146 .

Individual evidence

↑ ^a ^b ^c ^d Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. 2009, p. 29
↑ ^a ^b ^c P. S. Bullen, DS Mitrinović, Petar M. Vasić: Means and Their Inequalities. 1988, p. 77
↑ Alsina / Nelsen, op. Cit., Pp. 29-30

[CA-RBN-a-1] Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. 2009, p. 29

[PSB-DSM-PMV-a-2] P. S. Bullen, DS Mitrinović, Petar M. Vasić: Means and Their Inequalities. 1988, p. 77

[CA-RBN-b-3] Alsina / Nelsen, op. Cit., Pp. 29-30