相加平均・相乗平均・調和平均の関係

相加平均・相乗平均・調和平均の関係

\(k\in\mathbb{N}\;,\;0<x_{k}\)とする。

(1)

\[ \mu_{H}\left(x_{1},\cdots,x_{n}\right)=\frac{\mu_{G}^{\;n}\left(x_{1},\cdots,x_{n}\right)}{\mu_{A}\left(\frac{\left(\prod_{j=1}^{n}x_{k}\right)}{x_{1}},\cdots,\frac{\left(\prod_{j=1}^{n}x_{k}\right)}{x_{n}}\right)} \]

(2)

\[ \mu_{H}\left(x_{1},x_{2}\right)=\frac{\mu_{G}^{\;2}\left(x_{1},x_{2}\right)}{\mu_{A}\left(x_{1},x_{2}\right)} \]

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\(\mu_{A}\)は相加平均、\(\mu_{G}\)は相乗平均、\(\mu_{H}\)は調和平均。

(1)

\begin{align*} \mu_{H}\left(x_{1},\cdots,x_{n}\right) & =n\left(\sum_{k=1}^{n}\frac{1}{x_{k}}\right)^{-1}\\ & =n\left(\prod_{j=1}^{n}x_{k}\right)\left(\sum_{k=1}^{n}\frac{\left(\prod_{j=1}^{n}x_{k}\right)}{x_{k}}\right)^{-1}\\ & =\frac{\mu_{G}^{\;n}\left(x_{1},\cdots,x_{n}\right)}{\mu_{A}\left(\frac{\left(\prod_{j=1}^{n}x_{k}\right)}{x_{1}},\cdots,\frac{\left(\prod_{j=1}^{n}x_{k}\right)}{x_{n}}\right)} \end{align*}

(2)

(1)より、2変数のときは、
\[ \mu_{H}\left(x_{1},x_{2}\right)=\frac{\mu_{G}^{\;2}\left(x_{1},x_{2}\right)}{\mu_{A}\left(x_{1},x_{2}\right)} \]

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相加平均・相乗平均・調和平均の関係

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