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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Functions of finite $ \lambda $-type in several complex variables


Author: Robert O. Kujala
Journal: Trans. Amer. Math. Soc. 161 (1971), 327-358
MSC: Primary 32.12
MathSciNet review: 0281943
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Abstract: If $ \lambda :{{\bf {R}}^ + } \to {{\bf {R}}^ + }$ is continuous and increasing then a meromorphic function f on $ {C^k}$ is said to be of finite $ \lambda $-type if there are positive constants s, A, B and R such that $ {T_f}(r,s) \leqq A\lambda (Br)$ for all $ r > R$ where $ {T_f}(r,s)$ is the characteristic of f. It is shown that if $ \lambda (Br)/\lambda (r)$ is bounded for r sufficiently large and $ B > 1$, then every meromorphic function of finite $ \lambda $-type is the quotient of two entire functions of finite $ \lambda $-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 $ \nu $ on $ {C^k}$ with $ \nu ({\bf {0}}) = 0$ is the divisor of an entire function of finite $ \lambda $-type if and only if there are positive constants A, B and R such that

\begin{displaymath}\begin{array}{*{20}{c}} {{N_{\nu \vert\xi }}(r) \leqq A\lambd... ...ambda (Br){r^{ - p}} + A\lambda (Bs){s^{ - p}},} \\ \end{array}\end{displaymath}

for all $ r \geqq s > R$, all unit vectors $ \xi $ in $ {C^k}$, and all natural numbers p. Here $ \nu \vert\xi $ represents the lifting of the divisor $ \nu $ to the plane via the map $ z \mapsto z\xi $ and $ {N_{\nu \vert\xi }}$ is the valence function of that divisor.

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


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0281943-5
PII: S 0002-9947(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 $ \lambda $-type, order and type of a meromorphic function
Article copyright: © Copyright 1971 American Mathematical Society