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.

**[1]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[2]**Lars Hörmander,*An introduction to complex analysis in several variables*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR**0203075****[3]**H. Kneser,*Zur Theorie der gebrochenen Funktionen mehrer Verandlicher*, Jber. Deutsch. Math.-Verein.**48**(1938), 1-28.**[4]**E. Lindelöf,*Sur les fonctions entières d'ordre entier*, Ann. Sci. École Norm. Sup. (3)**22**(1905), 369-396.**[5]**L. A. Rubel and B. A. Taylor,*A fourier series method for meromorphic and entire functions*, Bull. Soc. Math. France**96**(1968), 53-96.**[6]**Wilhelm Stoll,*About entire and meromorphic functions of exponential type*, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 392–430. MR**0236410****[7]**E. C. Titchmarsh,*Han-shu lun*, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR**0197687**

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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