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

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

where

[math]\displaystyle{ C\gt 0,\quad \rho\gt 1\quad }[/math] and [math]\displaystyle{ \quad s\geq 0. }[/math]

Then [math]\displaystyle{ f }[/math] is of order [math]\displaystyle{ \rho }[/math] and has finite type.