Curious about intermediate mean between arithmetic and geometric, I noticed that all follow pattern
$M(\space f \space,\space x[1,2,...,n] \space,\space w[1,2,...,n] \space)=f^{-1}(\frac{w[1]\space*\space f(x[1]) \space+\space w[2]\space*\space f(x[2]) \space+\space ... \space+\space w[n]\space*\space f(x[n])}{w[1] \space+\space w[2] \space+\space ... \space+\space w[n]})$.
Arithmetic has $f=I=ln^0$, geometric, $f=ln=ln^1$. What would be intermediate of $ln^0$ and $ln^1$? I think it's $\sqrt{ln}$, ie functional square root (https://en.wikipedia.org/wiki/Functional_square_root). This falls on composite functions, more specifically finding $f$ such that $f\circ f=ln$.
There are a lot of possibilities but not all like taylor series with continuous derivatives, eg half-exponential (https://en.wikipedia.org/wiki/Half-exponential_function) with conditional branches and discontinuous second derivative. Comparing taylor series of $F$ and $f\circ f$ makes me think that maybe there is an official $f$ to be finded.
Even doesn't exists closed formula, can we find the taylor series up to a arbitrary degree of a half iteration function? If no, how to find a formula with at least the continuous derivatives of our preference?
For now I don't know about taylor series of $F=f\circ f$ but if can then may be converted to closed form with generator function. Since it is like $\sqrt{exp}$, $\sqrt{ln}$, ..., we can see the link https://en.wikipedia.org/wiki/Half-exponential_function with more attention.
We can understand it like this: as with $F=exp$, POSSIBLY if another $F$ is an increasing continuous bijector then there is $f$ such that $F=f\circ f$ and
$f(x) = \begin{cases} F^{-1}(f(F(x))) & x<A \\ g(x) & A\leq x\leq B \\F(g^ {-1}(x)) & B<x\leq C \\ F^{-1}(f(F(x))) & C<x \end{cases}$
where $g:[A,B]\to [B,C]$ is increasing continuous bijector. If we understand how this recursion works, we can also "define this possibility" better.
To be continued...
If you give me a better answer than what I'm thinking (in advance), I'll accept it as the solution.