(Note: This has been asked on Math SE, 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.
