mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence...

the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean of ⁠ n {\displaystyle n} ⁠ numbers is the nth...

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...

GA(x, y) is the geometric–arithmetic mean, A(x, y) is the arithmetic mean. Arithmetic–geometric mean Arithmetic–harmonic mean Mean Weisstein, Eric W...

Arithmetic-geometric mean Arithmetic-harmonic mean Cesàro mean Chisini mean Contraharmonic mean Elementary symmetric mean Geometric-harmonic mean Grand...

with the arithmetic mean, is the geometric mean to the power n. Thus the n-th harmonic mean is related to the n-th geometric and arithmetic means. The...

and OEIS: A223068 having the denominators. Given Eq. 3 and the arithmetic–geometric mean solution of the elliptic integral: K ( k ) = π 2 M ( 1 − k , 1...

numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). If p is a non-zero real number, and x 1 ,...

inequality of arithmetic and geometric means. For any real vector a = ( a 1 , … , a n ) {\displaystyle a=(a_{1},\dots ,a_{n})} define the "a-mean" [a] of positive...

y) denotes the arithmetic–geometric mean of x and y. It is obtained by repeatedly calculating the average (x + y)/2 (arithmetic mean) and x y {\textstyle...

repeatedly replaces two numbers by their arithmetic and geometric mean, in order to approximate their arithmetic-geometric mean. The version presented below is...

three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions...

}}x>0{\text{ and }}y>0.} Toyesh Prakash Sharma generalizes the arithmetic logarithmic geometric mean inequality for any n belongs to the whole number as x y...

different orders of magnitude. The limiting value is called the arithmetic-geometric mean of a {\displaystyle a} and b {\displaystyle b} , AGM ⁡ ( a , b...

from other types of means, such as geometric and harmonic. In addition to mathematics and statistics, the arithmetic mean is frequently used in economics...

term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is...

In statistics, the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean. Given a sample x = ( x 1 , x...

with 0 ≤ x ≤ ⁠1/2⁠. For different values of x, this Heinz mean interpolates between the arithmetic (x = 0) and geometric (x = 1/2) means such that for 0 < x < ⁠1/2⁠:...

Unlike most other elementary shapes, such as the circle and square, there is no closed-form expression for the perimeter of an ellipse. Throughout history...

usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric standard deviation is a multiplicative factor, and thus...

with the usual definition of the arithmetic–geometric mean for positive real numbers. See The Arithmetic-Geometric Mean of Gauss by David A. Cox. Milne...

and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different...

quarter period. It can be computed very efficiently in terms of the arithmetic–geometric mean: K ( k ) = π 2 agm ⁡ ( 1 , 1 − k 2 ) . {\displaystyle K(k)={\frac...

term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is...

quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean...

(c_{0},ib_{0})}}} M(w, z) is the arithmetic–geometric mean of w and z. At each step of the arithmetic–geometric mean iteration, the signs of zn arising...

laws. One of Gauss's first discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers. He discovered its relation...

with diameter p + q. Now the altitude represents the geometric mean and the radius the arithmetic mean of the two numbers. Since the altitude is always smaller...

the arithmetic-geometric mean inequality. Many important inequalities can be proved by the rearrangement inequality, such as the arithmetic mean – geometric...

Jensen's inequality, can be used to deduce inequalities such as the arithmetic–geometric mean inequality and Hölder's inequality. Let X {\displaystyle X} be...