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 1exp(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