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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 ''g''<sub>0</sub> = ''x'' and ''h''<sub>0</sub> = ''y'' and call it ''g''<sub>1</sub>, i.e. ''g''<sub>1</sub> is the [[square root]] of ''xy''. We also form the [[harmonic mean]] of ''x'' and ''y'' and call it ''h''<sub>1</sub>, i.e. ''h''<sub>1</sub> is the [[Multiplicative inverse|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 ''g''<sub>1</sub> taking the place of ''x'' and ''h''<sub>1</sub> taking the place of ''y''. In this way, two [[sequence]]s (''g''<sub>''n''</sub>) and (''h''<sub>''n''</sub>) are defined:
 
:<math>g_{n+1} = \sqrt{g_n h_n}</math>
 
and
 
:<math>h_{n+1} = \frac{2}{\frac{1}{g_n} + \frac{1}{h_n}}</math>
 
Both of these sequences [[limit (mathematics)|converge]] to the same number, which we call the '''geometric–harmonic mean''' M(''x'',&nbsp;''y'') of ''x'' and&nbsp;''y''. The geometric–harmonic mean is also designated as the '''harmonic–geometric mean'''.&nbsp;(cf. Wolfram MathWorld below.)
 
The existence of the limit can be proved by the means of [[Bolzano&ndash;Weierstrass theorem]] in a manner almost identical to the proof of existence of [[arithmetic–geometric mean]].
 
==Properties==
M(''x'',&nbsp;''y'') is a number between the geometric and harmonic mean of ''x'' and ''y''; in particular it is between ''x'' and ''y''. M(''x'',&nbsp;''y'') is also [[Homogeneous function|homogeneous]], i.e. if ''r''&nbsp;>&nbsp;0, then M(''rx'',&nbsp;''ry'') =&nbsp;''r'' M(''x'',&nbsp;''y'').
 
If AG(''x'', ''y'') is the [[arithmetic–geometric mean]], then we also have
 
:<math>M(x,y) = \frac{1}{AG(\frac{1}{x},\frac{1}{y})}</math>
 
==Inequalities==
We have the following inequality involving the Pythagorean means {''H'',&nbsp;''G'',&nbsp;''A''} and iterated Pythagorean means {''HG'',&nbsp;''HA'',&nbsp;''GA''}:
 
:<math>\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)</math>
 
where the iterated Pythagorean means have been identified with their parts {''H'',&nbsp;''G'',&nbsp;''A''} in progressing order:
 
* ''H''(''x'',&nbsp;''y'') is the harmonic mean,
* ''HG''(''x'',&nbsp;''y'') is the harmonic–geometric mean,
* ''G''(''x'',&nbsp;''y'') =&nbsp;''HA''(''x'',&nbsp;''y'') is the geometric mean (which is also the harmonic–arithmetic mean),
* ''GA''(''x'',&nbsp;''y'') is the geometric–arithmetic mean,
* ''A''(''x'',&nbsp;''y'') is the arithmetic mean.
 
==See also==
* [[Arithmetic–geometric mean]]
* [[Arithmetic–harmonic mean]]
* [[Mean]]
 
==External links==
* {{MathWorld|title=Harmonic-Geometric Mean|urlname=Harmonic-GeometricMean}}
 
{{DEFAULTSORT:Geometric-harmonic mean}}
[[Category:Means]]

Revision as of 09:54, 1 March 2014

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