Geometric–harmonic mean

Geometric–Harmonic Mean (GHM) is a mathematical concept that finds its application in various fields, including mathematics, statistics, and engineering. It represents a method of calculating the mean of two positive numbers, which combines the principles of both the geometric mean and the harmonic mean. The GHM is particularly useful in solving problems related to rates of growth, optimization, and in the analysis of algorithms.

Definition

The geometric–harmonic mean of two positive numbers a and b is defined as the limit of two sequences that are generated iteratively, starting from a and b. The sequences converge to the same limit, which is the GHM of a and b. Mathematically, it can be expressed as follows:

Let a₀ = a and b₀ = b, where a and b are the initial positive numbers. For n ≥ 1, define the sequences a_n and b_n iteratively by:

• a_n = ( a_n-1 + b_n-1 ) / 2 (arithmetic mean)
• b_n = √(a_n-1 * b_n-1) (geometric mean)

The GHM is the common limit of the sequences a_n and b_n as n approaches infinity.

Properties

The geometric–harmonic mean has several important properties:

• It is always less than or equal to the arithmetic mean of a and b.
• It is always greater than or equal to the harmonic mean of a and b.
• It is a special case of the generalized mean, with the power mean parameter being between 0 (geometric mean) and -1 (harmonic mean).

Applications

The GHM finds applications in various areas:

• In mathematics, it is used in the analysis of algorithms, particularly in the optimization of recursive algorithms.
• In physics, the GHM is applied in the calculation of resistances in parallel circuits and in the study of rates of cooling.
• In finance, it is used to calculate the compounded annual growth rate (CAGR) of investments.

Calculation

The calculation of the geometric–harmonic mean can be performed using iterative methods or by employing specialized mathematical software that can handle the convergence of sequences.

See Also

• Arithmetic mean
• Geometric mean
• Harmonic mean
• Generalized mean