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It is well-known that arithmetic and geometric mean are strongly related via logarithmic transformation, i.e. if we take arithmetic mean of logarithmic-transformed values we get the same as if we take logarithm of their geometric mean. With formula: $$\frac{\ln(x_1)+\cdots +\ln(x_n)}{n}=\ln \big( (x_1\cdots x_n)^{(1/n)} \big)$$ for all positive numbers $x_1, \ldots ,x_n$.

It is also known that both geometric and arithmetic mean belong to the same family of power mean (generalized mean) $$M_{\alpha}(x_1, \ldots ,x_n) = \Big( \frac{x_1^\alpha + \cdots + x_n^\alpha}{n} \Big)^{\frac{1}{\alpha}} $$ with $\alpha =1$ representing arithmetic mean and $\alpha=0$ (in limit) representing geometric mean.

Using this notation we can say that $M_1\circ \ln = \ln \circ M_0$ where $\circ$ represents the composition of functions.

My question is generalization of the case above. Let's say that we chose two real numbers $\alpha$ and $\beta$ and we are interested in function $f$ for which $M_\alpha\circ f = f \circ M_\beta$. Does such transformation function exist, is it unique (up to some constraint)?

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It seems that $f_{\beta,\alpha}(x)=x^{\beta/\alpha}$ do the trick for $\alpha<>0$.

$$M_\alpha\circ f_{\beta,\alpha} = \Big( \frac{(x_1^{{\beta/\alpha})^\alpha}+ \cdots + (x_n^{\beta/\alpha})^\alpha}{n} \Big)^{\frac{1}{\alpha}} = \Big( \frac{x_1^\beta + \cdots + x_n^\beta}{n} \Big)^{\frac{1}{\alpha}} $$ and $$f_{\beta,\alpha} \circ M_\beta= \Big(\Big( \frac{x_1^\beta+ \cdots + x_n^\beta}{n} \Big)^{\frac{1}{\beta}}\Big)^{\beta/\alpha}= \Big( \frac{x_1^\beta+ \cdots + x_n^\beta}{n} \Big)^{\frac{1}{\alpha}}$$

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