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)?
