Matsaev's theorem

Last updated June 06, 2023

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 $f(z)$ 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]

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References

↑ Mazaew, Wladimir Igorewitsch (1960). "On the growth of entire functions that admit a certain estimate from below". Soviet Math. Dokl. 1: 548–552.
↑ 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. doi:10.1007/978-3-0348-0516-2_12.

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[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. doi:10.1007/978-3-0348-0516-2_12.

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