Entire functions bounded outside a finite area (Q1066305)

Problem: Under what circumstances can it happen that for an entire function \(f(z)\) the 2-dimensional Lebesgue measure of \(\{z: |f(z)| > B\}\) is finite for some positive \(B\)? The authors answer this problem completely by proving that this can only happen, if \[ \lim \inf_{r\to \infty}\log \log \log M(r)/\log r\geq 2. \] An example shows that 2 can not be replaced by a larger number.