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Consider the equation

$$ f(f(x)) = \exp(\exp(x)) $$

Valid for all real $x$, $f(x) \neq \exp(x)$ (not identically equal everywhere), $f(x)$ is analytic and

$$ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ... $$

Where $a_1,a_2,a_3,\cdots > 0 $.

Find solutions for $f(x)$.

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Remarks :

I wonder about the behaviour of such an $f(z)$ on the complex plane.

Where does it satisfy the main equation NOT on the real axis? What about singularities? How does it grow in the imaginary direction? Where are its zeros etc.

But these are not the main question.

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  • $\begingroup$ Not the main questions, but it's easy to see that $f$ and $f'$ have no zeroes (differentiate both sides of the functional equation). also, such a $f$ would satisfy the same functional equation on whatever complex domain of definition of $f \circ f$, by the identity principle $\endgroup$ – Glougloubarbaki May 21 '18 at 15:57
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    $\begingroup$ Initial thoughts: If $f(x)$ is a polynomial (that is, there exists $m$ such that $a_t = 0$ for all $t \geq m$), then $f(f(x))$ is also a polynomial. It appears that the $n$-th derivative of $e^{e^x}$ will have the form $e^{e^x + x}(1 + b_1e^x + b_2e^{2x} + \dots + b_{n - 1}e^{(n - 1)x})$ where $b_1,\dots,b_{n - 1}$ are positive integers. This seems to imply none of the derivatives of $e^{e^x}$ has real zeros, whereas I don't see the same holding for a polynomial unless it is a constant... $\endgroup$ – theyaoster May 21 '18 at 16:11
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    $\begingroup$ @BrianYao it's also easy to see that $f$ cannot be a polynomial, since its iterates don't grow fast enough near infinity. (iterates of a degree $d$ polynomial grow like $z^{d^n}$, which is much slower than iterates of $\exp$) $\endgroup$ – Glougloubarbaki May 21 '18 at 16:28
  • $\begingroup$ Indeed, silly me. $\endgroup$ – theyaoster May 21 '18 at 16:31
  • $\begingroup$ @Glougloubarbaki : Well Yes and No. Whereever f is analytic and satisfies the equation and belongs to an analytic region that is dense in Some real interval Then , Yes tour statement about zero’s must be true. When the functional eq No longer holds or on branches all bets are off without further info id say. $\endgroup$ – mick May 21 '18 at 16:39

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