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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.

Matsaev's theorem

Let f ( z ) {\displaystyle f(z)} with z = r e i θ {\displaystyle z=re^{i\theta }} be an entire function which is bounded from below as follows

log ⁡ ( | f ( z ) | ) ≥ − C r ρ | sin ⁡ ( θ ) | s , {\displaystyle \log(|f(z)|)\geq -C{\frac {r^{\rho }}{|\sin(\theta )|^{s}}},}

where

C > 0 , ρ > 1 {\displaystyle C>0,\quad \rho >1\quad } and s ≥ 0. {\displaystyle \quad s\geq 0.}

Then f {\displaystyle f} is of order ρ {\displaystyle \rho } and has finite type.

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


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