In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers 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 interdependent sequences (gn) and (hn) are defined: and
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| - Geometric–harmonic mean (en)
- 기하 조화 평균 (ko)
- Średnia geometryczno-harmoniczna (pl)
- 几何-调和平均数 (zh)
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| - 수학에서 두 수 x, y의 기하 조화 평균(幾何調和平均) M(x, y)는 다음과 같이 정의된다. 우선 두 수 x, y의 기하 평균을 g1, 조화 평균을 h1라고 하자. 이후 g1과 h1을 x와 y 자리에 넣어 이 연산을 반복하면 두 수열을 얻게 된다. 이 두 수열은 같은 값으로 수렴하며, 이 수렴값을 x와 y의 기하 조화 평균이라 한다. M(x, y)은 x와 y의 기하 평균과 조화 평균의 사이값이다. r > 0에 대해, M(rx, ry) = r M(x, y)의 등식이 성립한다. (ko)
- Średnia geometryczno-harmoniczna dwóch liczb rzeczywistych dodatnich i – wspólna granica ciągów określonych rekurencyjnie: Granica ta istnieje dla dowolnych rzeczywistych dodatnich, a dowód tego faktu jest analogiczny do dowodu istnienia średniej arytmetyczno-geometrycznej. (pl)
- 两个正实数x和y的几何-调和平均数（英語：Geometric–harmonic mean）是一种二元平均数。 (zh)
- In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers 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 interdependent sequences (gn) and (hn) are defined: and (en)
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| - Harmonic-Geometric Mean (en)
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| - Harmonic-GeometricMean (en)
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| - In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers 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 interdependent sequences (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. (en)
- 수학에서 두 수 x, y의 기하 조화 평균(幾何調和平均) M(x, y)는 다음과 같이 정의된다. 우선 두 수 x, y의 기하 평균을 g1, 조화 평균을 h1라고 하자. 이후 g1과 h1을 x와 y 자리에 넣어 이 연산을 반복하면 두 수열을 얻게 된다. 이 두 수열은 같은 값으로 수렴하며, 이 수렴값을 x와 y의 기하 조화 평균이라 한다. M(x, y)은 x와 y의 기하 평균과 조화 평균의 사이값이다. r > 0에 대해, M(rx, ry) = r M(x, y)의 등식이 성립한다. (ko)
- Średnia geometryczno-harmoniczna dwóch liczb rzeczywistych dodatnich i – wspólna granica ciągów określonych rekurencyjnie: Granica ta istnieje dla dowolnych rzeczywistych dodatnich, a dowód tego faktu jest analogiczny do dowodu istnienia średniej arytmetyczno-geometrycznej. (pl)
- 两个正实数x和y的几何-调和平均数（英語：Geometric–harmonic mean）是一种二元平均数。 (zh)
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