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]

References

1. Mazaew, Wladimir Igorewitsch (1960). "On the growth of entire functions that admit a certain estimate from below". Soviet Math. Dokl. 1: 548–552.
2. Kheyfits, A.I. (2013). "Growth of Schrödingerian Subharmonic Functions Admitting Certain Lower Bounds". Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications. Vol. 229. Basel: Birkhäuser. pp. 215–231. doi:10.1007/978-3-0348-0516-2_12. ISBN   978-3-0348-0515-5.