(06/06/2022, 02:54 AM)Catullus Wrote:(06/05/2022, 11:36 PM)JmsNxn Wrote: This sounds terrifyingly hard. Even showing something like this converges sounds terrifyingly hard, lol. Not sure how much luck you'll have, but keep me posted if you think of anything, lol.Like how hard exactly? Exp time? Exp-mean(2,3) ~ 2.419. Exp-mean(2,4) = ln(16)/W0(ln(16)) ~ 2.745. Does anyone know of any series for this, like a series for exp-meaan(x,2)?
Oh, I apologize.
What I mean, is I can bet this probably converges! But I have no idea how to do it!
I mean, this looks like something that would probably converge, and I don't doubt it converges. PROVING it converges sounds impossible
, at least, that's what I meant.I believe you'd have more luck looking at arithmetic/geometric means for general functions. Look at if \(f,g\) are reasonable functions does a similar process converge. This would mean, looking at:
\[
\begin{align}
F&=f^{-1}(f(a,b)*f(b,a),b)\\
G&=f^{-1}(a,f(a,b)*f(b,a))\\
&...\\
\end{align}
\]
What I mean to say, is that you could probably find literature on similar cases for general functions; especially how you can iterate and mix arithmetic and geometric means (this was a proposed solution of semi-operators on this forum, the trouble being it didn't satisfy the Goodstein equation). I would look into the general theory of iterated arithmetic/geometric means; which does exist, and view how they show convergence. I know there is research on \(f(a,b)\)'s mean, and so on and so forth. It brushes path with iteration theory. I apologize but I'm not the one to ask. I'd google it, but I'm not even sure what to google...
EDIT:
I kinda went down a rabbit hole, but this seems a good place to start:
https://www.jstor.org/stable/pdf/1967353.pdf
You'd have to do much of the techniques here, but with general functions as opposed to the arithmetic/geometric mean stuff.