(*Note:* This has been [asked on Math SE](https://math.stackexchange.com/q/3783159/42969), but without an answer after almost two years and one offered bounty.)


For an entire function $f$ let $M(r,f)=\max\_{|z|=r}|f(z)|$ be its maximum modulus function. $M(r, f)$ does not change


* if $f$ is replaced by $e^{i\varphi}f(e^{i\theta}z)$,
* if $f$ is replaced by by $\overline{f(\overline z)}$,
* or combinations thereof.


My question is:



> 
> What can be said about entire functions $f$ and $g$ which satisfy $M(r, f) = M(r, g)$ for *all* $r > 0$?
> 
> 
> 


Are $f$ and $g$ necessarily related by the above listed transformations?


Some thoughts:


* If $f(0) = 0$ with multiplicity $k$ at the origin then $M(r, f) \sim r^k$ for $r \to 0$, so that $g(0) = 0$ with the same multiplicity. We can divide both functions by $z^k$ and therefore assume that $|f(0)| = |g(0)| \ne 0$. After multiplying both functions with suitable factors $e^{i\varphi}$ we can assume that $f(0) = g(0) = 1$.
* If $f(z) = 1 + a\_m z^m + \cdots$ at the origin with $a\_m \ne 0$ then $M(r, f) \sim 1 + |a\_m| r^m$ for $r \to 0$, so $g(z) = 1 + b\_m z^m + \cdots$ with $|b\_m| = |a\_m|$. After suitable rotations $z \to e^{i\theta}z$ in the argument we can assume that $a\_m = b\_m > 0$.


After this normalization the question is whether necessarily $g = f$ or $g(z) = \overline{f(\overline z)}$, or if no such conclusion can be drawn.



Are entire functions “essentially” determined by their maximum modulus function?
This is a classical problem, but only partial results are available:


MR3155684
Hayman, W. K.; Tyler, T. F.; White, D. J.
The Blumenthal conjecture, in [*Complex Analysis and Dynamical Systems V*](https://web.archive.org/web/20220428174502/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.364.3299&rep=rep1&type=pdf), 149–157.


On the latest results see:


MR4348902
Evdoridou, Vasiliki; Pardo-Simón, Leticia; Sixsmith, David J.
[On a result of Hayman concerning the maximum modulus set](http://oro.open.ac.uk/78421/1/Evdoridou2021_Article_OnAResultOfHaymanConcerningThe.pdf).
*Comput. Methods Funct. Theory* 21 (2021), no. 4, 779–795,


and references there.


