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In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
Proof without words of the AM–GM inequality: PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ. Visual proof that (x + y) 2 ≥ 4xy. Taking square roots and dividing by two gives the AM ...
By weighted AM-GM on , + ... and by the formula for geometric series ... Online e-book in PDF format. This page was last ...
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians [ 1 ] because of their importance in geometry and music.
Template: QM AM GM HM inequality visual proof.svg. ... Print/export Download as PDF; Printable version
In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we form the geometric mean of g 0 = x and h 0 = y and call it g 1, i.e. g 1 is the square root of xy.
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Proof without words of the AM–GM inequality: PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ.