A conjectured uniqueness criteria for analytic tetration

[Vladimir Reshetnikov]

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 \( a \) be a fixed real number in the interval \( 1 < a < e^{1/e} \). There is a unique function \( f(z) \) of a complex variable \( z \), defined on the complex half-plane \( \Re(z) > -2 \), and satisfying all of the following conditions:

* \( f(0) = 1 \).
* The identity \( f(z+1) = a^{f(z)} \) holds for all complex \( z \) in its domain (together with the first condition, it implies that \( f(n) = {^n a} \) for all \( n \in \mathbb N \)).
* For real \( x > -2, \, f(x) \) is a continuous real-valued function, and its derivative \( f'(x) \) is a completely monotone function (this condition alone implies that the function \( f(x) \) is real-analytic for \( x > -2 \)).
* The function \( f(z) \) 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.

[sheldonison]

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 \( a \) be a fixed real number in the interval \( 1 < a < e^{1/e} \). There is a unique function \( f(z) \) of a complex variable \( z \), defined on the complex half-plane \( \Re(z) > -2 \), and satisfying all of the following conditions:

* \( f(0) = 1 \).
* The identity \( f(z+1) = a^{f(z)} \) holds for all complex \( z \) in its domain (together with the first condition, it implies that \( f(n) = {^n a} \) for all \( n \in \mathbb N \)).
* For real \( x > -2, \, f(x) \) is a continuous real-valued function, and its derivative \( f'(x) \) is a completely monotone function (this condition alone implies that the function \( f(x) \) is real-analytic for \( x > -2 \)).
* The function \( f(z) \) 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?

[tommy1729]

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

[sheldonison]

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
\( S(z)= L+ \sum -a_n \lambda ^{nz} \;\;\lambda<1 \)
\( S(z)\approx L - \lambda ^{z} \)

For simplicity, lets look at the closely related function
\( f(z) = -\exp(-z) \)
and compare the perfectly behaved derivatives of f(z) as compared with the alternative solution
\( g(z) = -\exp(-z-\theta(z))\;\; \) 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)

[tommy1729]

If you can show me existance i can probably get a proof of uniqueness. Or at least arguments.

Regards

Tommy1729

[Vladimir Reshetnikov]

Related conjectures posted at MathOverflow: http://mathoverflow.net/q/259278/9550

[Vladimir Reshetnikov]

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

[JmsNxn]

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.

[JmsNxn]

http://mathoverflow.net/questions/260298...-equations

So I asked on Mathoverflow if \( \lambda^x \phi(x) \) is completely monotone where \( \phi \) is 1-periodic, must \( \phi \) 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.

[Vladimir Reshetnikov]

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.