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In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which ...
Then the length of GF can be calculated to be the harmonic mean, CF to be the geometric mean, DE to be the arithmetic mean, and CE to be the quadratic mean. The inequalities then follow easily by the Pythagorean theorem. Comparison of harmonic, geometric, arithmetic, quadratic and other mean values of two positive real numbers and
For example, the geometric mean of 2 and 3 is 2.45, while their arithmetic mean is 2.5. In particular, this means that when a set of non-identical numbers is subjected to a mean-preserving spread — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean ...
The geometric mean of two positive numbers is never greater than the arithmetic mean. [3] So the geometric means are an increasing sequence g 0 ≤ g 1 ≤ g 2 ≤ ...; the arithmetic means are a decreasing sequence a 0 ≥ a 1 ≥ a 2 ≥ ...; and g n ≤ M(x, y) ≤ a n for any n. These are strict inequalities if x ≠ y.
In mathematics, generalised means (or power mean or Hölder mean from Otto Hölder) [1] are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means ( arithmetic , geometric , and harmonic means ).
This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The proof follows from the arithmetic–geometric mean inequality , AM ≤ max {\displaystyle \operatorname {AM} \leq \max } , and reciprocal duality ( min {\displaystyle \min } and max {\displaystyle \max ...
Maclaurin's inequality is the following chain of inequalities: with equality if and only if all the are equal. For n = 2 {\displaystyle n=2} , this gives the usual inequality of arithmetic and geometric means of two non-negative numbers.
In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. Preliminary definitions [ edit ]