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 A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov Junior Fellow Posts: 12 Threads: 3 Joined: Dec 2011 10/30/2016, 11:02 PM (This post was last modified: 10/30/2016, 11:08 PM by 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 Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 11/01/2016, 02:08 AM (This post was last modified: 11/01/2016, 07:46 PM by sheldonison.) (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/...-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

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