mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the geometric–harmonic mean M(''x'', ''y'') of two positive

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s ''x'' and ''y'' is defined as follows: we form the

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...

of ''g''₀ = ''x'' and ''h''₀ = ''y'' and call it ''g''₁, i.e. ''g''₁ is the

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

of ''xy''. We also form the

harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean ...

of ''x'' and ''y'' and call it ''h''₁, i.e. ''h''₁ is the

reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...

of the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

of the reciprocals of ''x'' and ''y''. These may be done sequentially (in any order) or simultaneously. Now we can iterate this operation with ''g''₁ taking the place of ''x'' and ''h''₁ taking the place of ''y''. In this way, two interdependent

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

s (''g''_''n'') and (''h''_''n'') are defined: :

g_ = \sqrt

and :

h_ = \frac

Both of these sequences

converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) In mathematics, a limit is the value that a function (or sequence) app ...

to the same number, which we call the geometric–harmonic mean M(''x'', ''y'') of ''x'' and ''y''. The geometric–harmonic mean is also designated as the harmonic–geometric mean. (cf. Wolfram MathWorld below.) The existence of the limit can be proved by the means of Bolzano–Weierstrass theorem in a manner almost identical to the proof of existence of

arithmetic–geometric mean In mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms f ...

Properties

M(''x'', ''y'') is a number between the geometric and harmonic mean of ''x'' and ''y''; in particular it is between ''x'' and ''y''. M(''x'', ''y'') is also

homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...

, i.e. if ''r'' > 0, then M(''rx'', ''ry'') = ''r'' M(''x'', ''y''). If AG(''x'', ''y'') is the

, then we also have :

M(x,y) = \frac

Inequalities

We have the following inequality involving the Pythagorean means and iterated Pythagorean means : :

\min(x,y) \leq H(x,y) \leq HG(x,y) \leq G(x,y) \leq GA(x,y) \leq A(x,y) \leq \max(x,y)

where the iterated Pythagorean means have been identified with their parts in progressing order: * ''H''(''x'', ''y'') is the harmonic mean, * ''HG''(''x'', ''y'') is the harmonic–geometric mean, * ''G''(''x'', ''y'') = ''HA''(''x'', ''y'') is the geometric mean (which is also the harmonic–arithmetic mean), * ''GA''(''x'', ''y'') is the geometric–arithmetic mean, * ''A''(''x'', ''y'') is the arithmetic mean.

External links

* {{DEFAULTSORT:Geometric-harmonic mean Means

Properties

Inequalities

See also

External links