Matsaev's theorem explained

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

	r^\rho
	\|\sin(\theta)\|^s

where

C>0, \rho>1

and

s\geq0.

Then

is of order

\rho

and has finite type.^[2]

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