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.