Matsaev's theorem is a theorem from

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

, which characterizes the order and type of an

entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...

. The theorem was proven in 1960 by Vladimir Igorevich Matsaev.

Matsaev's theorem

Let

f(z)

with

z=re^

be an entire function which is bounded from below as follows :

\log(, f(z), )\geq -C\frac,

where :

C>0,\quad \rho>1\quad

and

\quad s\geq 0.

Then

f

is of order

\rho

and has finite type.

References

Theorems in complex analysis {{improve categories, date=June 2023