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 be a fixed real number in the interval . There is a unique function of a complex variable , defined on the complex halfplane , 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 realvalued function, and its derivative is a completely monotone function (this condition alone implies that the function is realanalytic 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 counterexamples 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: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/completemonotonicityofasequencerelatedtotetration 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 bohrmollerup. 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 ?? 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 1cyclic 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/istheexponentialfunctionthesolesolutiontotheseequations 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/istheexponentialfunctionthesolesolutiontotheseequations So I asked on Mathoverflow if is completely monotone where is 1periodic, 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. 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 semiaxis) is always analytic on that interval. 