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(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.

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1 Answer 1

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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, 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. Comput. Methods Funct. Theory 21 (2021), no. 4, 779–795,

and references there.

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