Matsaev's theorem

Matsaev's theorem is a theorem from complex analysis, which characterizes the order and type of an entire function.

The theorem was proven in 1960 by Vladimir Igorevich Matsaev.

Matsaev's theorem

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

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

where

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

Then $f$ is of order $\rho$ and has finite type.