Geometric-harmonic mean
Encyclopedia
In mathematics
, the geometric-harmonic mean M(x, y) of two positive real number
s x and y is defined as follows: we form the geometric mean
of g0 = x and h0 = y and call it g1, i.e. g1 is the square root
of xy. We also form the harmonic mean
of x and y and call it h1, i.e. h1 is the reciprocal
of the arithmetic mean
of the reciprocals of x and y. These may be done sequentially (in any order) or simultaneously.
Now we can iterate this operation with g1 taking the place of x and h1 taking the place of y. In this way, two sequence
s (gn) and (hn) are defined:
and
Both of these sequences converge
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.
, i.e. if r > 0, then M(rx, ry) = r M(x, y).
If AG(x, y) is the arithmetic-geometric mean, then we also have
where the iterated Pythagorean means have been identified with their parts {H, G, A} in progressing order:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the geometric-harmonic mean M(x, y) of two positive real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s x and y is defined as follows: we form the geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
of g0 = x and h0 = y and call it g1, i.e. g1 is the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
of xy. We also form the harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
of x and y and call it h1, i.e. h1 is the reciprocal
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
of the reciprocals of x and y. These may be done sequentially (in any order) or simultaneously.
Now we can iterate this operation with g1 taking the place of x and h1 taking the place of y. In this way, two sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s (gn) and (hn) are defined:
and
Both of these sequences converge
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
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.
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 homogeneousHomogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
, i.e. if r > 0, then M(rx, ry) = r M(x, y).
If AG(x, y) is the arithmetic-geometric mean, then we also have
Inequalities
We have the following inequality involving the Pythagorean means {H, G, A} and iterated Pythagorean means {HG, HA, GA}:where the iterated Pythagorean means have been identified with their parts {H, G, A} 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.