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 $z=r{e}^{i\theta }$ be an entire function which is bounded from below as follows

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

where

$C>0,\rho >1$ and $s\ge 0.$

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