Functions of finite -type in several complex variables

Author:
Robert O. Kujala

Journal:
Trans. Amer. Math. Soc. **161** (1971), 327-358

MSC:
Primary 32.12

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281943-5

MathSciNet review:
0281943

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Abstract: If is continuous and increasing then a meromorphic function *f* on is said to be of finite -type if there are positive constants *s, A, B* and *R* such that for all where is the characteristic of *f*. It is shown that if is bounded for *r* sufficiently large and , then every meromorphic function of finite -type is the quotient of two entire functions of finite -type.

This theorem is the result of a careful and detailed analysis of the relation between the growth of a function and the growth of its divisors. The central fact developed in this connection is: A nonnegative divisor on with is the divisor of an entire function of finite -type if and only if there are positive constants *A, B* and *R* such that

*p*. Here represents the lifting of the divisor to the plane via the map and is the valence function of that divisor.

Analogous facts for functions of zero -type are also presented.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281943-5

Keywords:
Several complex variables,
meromorphic function,
holomorphic function,
characteristic of a meromorphic function,
value-distribution theory,
divisors of a function,
plurisubharmonic functions,
Fourier coefficients of a meromorphic function,
growth functions,
growth indices,
finite -type,
order and type of a meromorphic function

Article copyright:
© Copyright 1971
American Mathematical Society