On the area of the region where an entire function is greater than one

Abstract:

Using Carleman’s inequality, we prove that if $f$ is entire and of finite order $\rho \geq 1$, then \[ \lim sup\limits _{r \to \infty } \frac {{A(r)}}{{{r^2}}} \geq \frac {\pi }{{2\rho }},\] where $A(r)$ is the area of the region $\{ z:|f(z)| \geq 1\;{\text {and}}\;|z| \leq r\}$.