Claims

../

1960_12_01_hayman: Hayman's 1960 theorem: an entire function with log M(r,f) = O((log r)^2) tends to infinity along almost every ray, so a ray gives a path with l(r) = r; the yes-instances of Problem 1115 for that growth class.

1979_01_01_goldberg_eremenko: Gol'dberg and Eremenko's 1979 Theorems 1 and 2: entire functions of order zero growing barely above Hayman's threshold, and of every finite order, with no path on which f tends to infinity having length O(r) inside the disc of radius r.

1980_02_01_toppila: Toppila's 1980 note: for every increasing phi tending to infinity, an entire function f with log M(r,f) = O(phi(r)(log r)^2) such that every path where f tends to infinity has limsup l(r)/r infinite; an independent disproof.

---