Matsaev's theorem

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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]

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.^[2]

References

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