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A conjectured uniqueness criteria for analytic tetration
#1
After some study of different approaches to an extension of tetration to fractional or complex heights, and many numeric experiments, I came to the following conjecture, that I am currently trying to prove:

Let be a fixed real number in the interval . There is a unique function of a complex variable , defined on the complex half-plane , and satisfying all of the following conditions:

* .
* The identity holds for all complex in its domain (together with the first condition, it implies that for all ).
* For real is a continuous real-valued function, and its derivative is a completely monotone function (this condition alone implies that the function is real-analytic for ).
* The function is holomorphic on its domain.

Please kindly let me know if this conjecture has been already proved, or if you know any counter-examples to it, or if you have any ideas about how to approach to proving it.
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#2
(10/30/2016, 11:02 PM)Vladimir Reshetnikov Wrote: After some study of different approaches to an extension of tetration to fractional or complex heights, and many numeric experiments, I came to the following conjecture, that I am currently trying to prove:

Let be a fixed real number in the interval . There is a unique function of a complex variable , defined on the complex half-plane , and satisfying all of the following conditions:

* .
* The identity holds for all complex in its domain (together with the first condition, it implies that for all ).
* For real is a continuous real-valued function, and its derivative is a completely monotone function (this condition alone implies that the function is real-analytic for ).
* The function is holomorphic on its domain.

Please kindly let me know if this conjecture has been already proved, or if you know any counter-examples to it, or if you have any ideas about how to approach to proving it.
There is a proof framework for how to show that the standard solution from the Schröder equation is completely monotonic. The framework only applies to tetration bases 1<b<exp(1/e) and does not apply to Kneser's solution for bases>exp(1/e), which is a different analytic function.
http://math.stackexchange.com/questions/...-tetration

The conjecture would be that the completely monotonic criteria is sufficient for uniqueness as well; that there are no other completely monotonic solutions. It looks like there is a lot of theorems about completely monotone functions but I am not familiar with this area of study. Can you suggest a reference? I found https://en.wikipedia.org/wiki/Bernstein%..._functions

Also, is the inverse of a completely monotone function also completely monotone? no, that doesn't work. So what are the requirements for the slog for bases<exp(1/e) given that sexp is completely monotonic?
- Sheldon
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#3
Why do people think it is true ??
What arguments are used ?

Only boundedness seems to give uniqueness so far ?
Like bohr-mollerup.

I do not see the properties giving Uniqueness unless if those boundedness are a consequence ...

Regards

Tommy1729
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#4
(11/30/2016, 01:26 AM)tommy1729 Wrote: Why do people think it is true ??
What arguments are used ?

Only boundedness seems to give uniqueness so far ?
Like bohr-mollerup.

I do not see the properties giving Uniqueness unless if those boundedness are a consequence ...

Regards

Tommy1729

From my mathstack answer, the exact solution for S(z) for bases 1<b<exp(1/e),
has the following form. Consider the increasingly good approximation z goes to infinity



For simplicity, lets look at the closely related function

and compare the perfectly behaved derivatives of f(z) as compared with the alternative solution
where theta is 1-cyclic

The conjecture is g(z) can be shown to be not fully monotonic unless theta(z) is a constant. And then with a little bit of work, this can be used to show that S(z) is the unique completely monotonic solution to the Op's problem for bases b<exp(1/e)
- Sheldon
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#5
If you can show me existance i can probably get a proof of uniqueness. Or at least arguments.

Regards

Tommy1729
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#6
Related conjectures posted at MathOverflow: http://mathoverflow.net/q/259278/9550
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#7
I found that this conjecture was already proposed on this forum several years ago: http://math.eretrandre.org/tetrationforu...41#pid5941 and http://math.eretrandre.org/tetrationforu...237#pid237
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#8
So I feel like the solution to tetration whose derivative is completely monotone is definitely unique. In attempts at solving this, the biggest obstacle I found, one I avoided and just assumed, is that the exponential function is the only completely monotone solution to some multiplicative equations. I posted the question on MO
http://mathoverflow.net/questions/260298...-equations
I think if we have this complete monotonicity will follow from this. This is mostly because of Sheldon's proof that the schroder tetration is completely monotone.
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#9
http://mathoverflow.net/questions/260298...-equations

So I asked on Mathoverflow if  is completely monotone where is 1-periodic, must be a constant?

The beautiful answer is yes, which I think further cements the fact that bounded tetration is unique if it is completely monotone.
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#10
Nice, thanks! I would like to mention that a smooth and completely monotonic function on an open interval (or semi-axis) is always analytic on that interval.
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