Let $f$ be an entire function, and $$M_f(r)=\max_{|z|\leq r}|f(z)|$$ denotes its maximum modulus on the circle centered at the origin with radius $r>0$.
It's clear that for any entire functions $f(z)$ and $g(z)=e^{i\varphi}f(e^{i\theta}z)$, $M_f\equiv M_g$, where $\varphi$ and $\theta$ are real constants.
For $f(z)$ and $h(z)=\overline{f(\overline z)}$, we also have $M_f\equiv M_h$.
My question is, for any entire functions $f(z), g(z)$ that satisfying $M_f\equiv M_g$, could we always transform $f$ into $g$ by the rotation and reflection(or some composition of both) as above? If not, is there any other universal transforms that keeps $M_f$?
Best Answer
It is an old classical conjecture of Blumenthal. The most recent results are here:
Hayman, W. K.; Tyler, T. F.; White, D. J. The Blumenthal conjecture. Complex analysis and dynamical systems V, 149–157, Contemp. Math., 591, Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 2013.