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.^[1]

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]