A conjectured uniqueness criteria for analytic tetration

$Tetration Forum$

A conjectured uniqueness criteria for analytic tetration - Printable Version

+- Tetration Forum (https://math.eretrandre.org/tetrationforum)
+-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)
+--- Thread: A conjectured uniqueness criteria for analytic tetration (/showthread.php?tid=1102)

Pages: 1 2

A conjectured uniqueness criteria for analytic tetration - Vladimir Reshetnikov - 10/30/2016

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.

RE: A conjectured uniqueness criteria for analytic tetration - sheldonison - 11/01/2016

(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 $a$ 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/1987944/complete-monotonicity-of-a-sequence-related-to-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%27s_theorem_on_monotone_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?

RE: A conjectured uniqueness criteria for analytic tetration - tommy1729 - 11/30/2016

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

RE: A conjectured uniqueness criteria for analytic tetration - sheldonison - 12/01/2016

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

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)

RE: A conjectured uniqueness criteria for analytic tetration - tommy1729 - 12/07/2016

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

Regards

Tommy1729

RE: A conjectured uniqueness criteria for analytic tetration - Vladimir Reshetnikov - 01/11/2017

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

RE: A conjectured uniqueness criteria for analytic tetration - Vladimir Reshetnikov - 01/15/2017

I found that this conjecture was already proposed on this forum several years ago: http://math.eretrandre.org/tetrationforum/showthread.php?tid=503&pid=5941#pid5941 and http://math.eretrandre.org/tetrationforum/showthread.php?tid=37&pid=237#pid237

RE: A conjectured uniqueness criteria for analytic tetration - JmsNxn - 01/23/2017

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/is-the-exponential-function-the-sole-solution-to-these-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.

RE: A conjectured uniqueness criteria for analytic tetration - JmsNxn - 01/24/2017

http://mathoverflow.net/questions/260298/is-the-exponential-function-the-sole-solution-to-these-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.

RE: A conjectured uniqueness criteria for analytic tetration - Vladimir Reshetnikov - 01/25/2017

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.