This is a classical problem, but only partial results are available:

MR3155684 Hayman, W. K.; Tyler, T. F.; White, D. J. The Blumenthal conjecture. 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.

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.