Matsaev's theorem

Let ${\displaystyle f(z)}$ with ${\displaystyle z=re^{i\theta }}$ be an entire function which is bounded from below as follows

${\displaystyle \log(|f(z)|)\geq -C{\frac {r^{\rho }}{|\sin(\theta )|^{s}}},}$

where

${\displaystyle C>0,\quad \rho >1\quad }$ and ${\displaystyle \quad s\geq 0.}$

Then ${\displaystyle f}$ is of order ${\displaystyle \rho }$ and has finite type.^[2]