If there are entire $G_k$s such that $f=\exp\circ\exp\circ\cdots \circ\exp\circ G_k$ ($k$ times), must $f$ be constant?

Question

I am a French guest and I hope that my English isn't too bad...

So here is my issue: I consider an entire function $f$ which satisfies the following property for each complex number $z\in \mathbb{C}$:

$\forall ~ k \in \mathbb{N}^*$, there exists an entire function $G_k$ that satisfies $$f(z)=\exp_k(G_k(z))$$ where $\exp_k$ denotes $\exp \circ \exp \circ ...\circ \exp$, $k$-times.

In other words, I can take (as many times as I wish) the $\log$ of my function $f$, and it will always give an entire function that doesn't vanish on $\mathbb{C}$.

Is $f$ a constant? (surely different from $0$...)

Thanks everyone!

(PS: This forum is so cool!)

Answer 1

I think that the concept of 'order of an entire function' can help you to get an answer.
(See definition in W. Rudin : Exercise 2, Ch. 15 of ; or alternatively here : Wiki - Entire function )

Namely, your entire function $f$ is clearly of infinite order (well, at least for k>1), while constant functions -like polynomials- are entire functions of order zero ...

NB: of course I'm not considering the trivial case where the $G_k$ are constant (and non-zero), in which case $f$ would be constant as well ...