To clarify this a bit more.
We know that this function f(x) = exp^(1/2)[ln^(1/2)[x]+1] - x must be smaller than x and greater than 0 and we know that if we iterate f(x)+x 'x' times we get (about) exp^(1/2)[x] which grows faster than any polynomial and slower than any exponential ( any base ).
( exp^(1/2) is like a superfunction here )
An asymptotic expression for f(x) in terms of elementary functions and/or integrals and/or number theoretical functions would amaze everyone because it probably does not exist.
Hence the next expression we logically look for is also an iteration , just like most expressions concerning dynamical systems and tetration.
For instance (and by lack of replies of other solutions )
Let g(x) = f(x) + x = exp^(1/2)[ln^(1/2)[x]+1]
Then we state g(x) is O( x th iteration of h(x) ) where O is big O notation.
( by lack of good results I am forced to use big O at the moment )
Let g(0) = h(0) = 2.
Although I was not able to find a good h function , I considered another function called H(x).
the point of h and H is that they must have simple forms but nonconventional superfunctions ; nonconvential growth rates.
( see beginning of this post )
The H(x) I might have found ( needs confirmation and proofs etc ) comes as a special case of a generalization of something I call " a transition ".
Consider H(x,a,b) = x(1+1/ln(x)^(a+(b/x)))
For 1=<a=<5 The superfunction of H(x,a,b) is O(exp(U x^I + O)) where U,I,O depend mainly on a.
For a>=7 however the superfunction of H(x,a,b) is O(J B^x + L) where J,B,L mainly depend on a.
However near a = 5.95 it appears the superfunction behaves somewhere in between and the value of b matters more.
This is " the transition " and for a near 5.95 it is unclear if we suddenly go from O(exp(U x^I + O)) towards O(J B^x + L) or there is growth rate inbetween.
I must say I did not investigate the superfunction of this yet , nor its fractals. MAYBE WE NEED TO INVESTIGATE THE FIXPOINT AT INFINITY ??
Also I do not fully understand these " transitions " yet , e.g. are there functions with multiple transistions ?
Let H(x) = x(1+1/ln(x)^(a*+(b*/x))) where a* is about 5.95 and b* is 3.1415.
Then this H(x) might be a function that lies between polynomials and exponentiations in a nontrivial ( not asymptotic to an elementary function ) way.
Thus H(x) might be a first step to h(x).
And this might be the road to an answer.
However like I said , alot of mights and maybe's and guesses.
Hoping for progress.
Regards
Tommy1729
We know that this function f(x) = exp^(1/2)[ln^(1/2)[x]+1] - x must be smaller than x and greater than 0 and we know that if we iterate f(x)+x 'x' times we get (about) exp^(1/2)[x] which grows faster than any polynomial and slower than any exponential ( any base ).
( exp^(1/2) is like a superfunction here )
An asymptotic expression for f(x) in terms of elementary functions and/or integrals and/or number theoretical functions would amaze everyone because it probably does not exist.
Hence the next expression we logically look for is also an iteration , just like most expressions concerning dynamical systems and tetration.
For instance (and by lack of replies of other solutions )
Let g(x) = f(x) + x = exp^(1/2)[ln^(1/2)[x]+1]
Then we state g(x) is O( x th iteration of h(x) ) where O is big O notation.
( by lack of good results I am forced to use big O at the moment )
Let g(0) = h(0) = 2.
Although I was not able to find a good h function , I considered another function called H(x).
the point of h and H is that they must have simple forms but nonconventional superfunctions ; nonconvential growth rates.
( see beginning of this post )
The H(x) I might have found ( needs confirmation and proofs etc ) comes as a special case of a generalization of something I call " a transition ".
Consider H(x,a,b) = x(1+1/ln(x)^(a+(b/x)))
For 1=<a=<5 The superfunction of H(x,a,b) is O(exp(U x^I + O)) where U,I,O depend mainly on a.
For a>=7 however the superfunction of H(x,a,b) is O(J B^x + L) where J,B,L mainly depend on a.
However near a = 5.95 it appears the superfunction behaves somewhere in between and the value of b matters more.
This is " the transition " and for a near 5.95 it is unclear if we suddenly go from O(exp(U x^I + O)) towards O(J B^x + L) or there is growth rate inbetween.
I must say I did not investigate the superfunction of this yet , nor its fractals. MAYBE WE NEED TO INVESTIGATE THE FIXPOINT AT INFINITY ??
Also I do not fully understand these " transitions " yet , e.g. are there functions with multiple transistions ?
Let H(x) = x(1+1/ln(x)^(a*+(b*/x))) where a* is about 5.95 and b* is 3.1415.
Then this H(x) might be a function that lies between polynomials and exponentiations in a nontrivial ( not asymptotic to an elementary function ) way.
Thus H(x) might be a first step to h(x).
And this might be the road to an answer.
However like I said , alot of mights and maybe's and guesses.
Hoping for progress.
Regards
Tommy1729