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

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 \[ \begin {array}{*{20}{c}} {{N_{\nu |\xi }}(r) \leqq A\lambda (Br),} \\ {\left | {\frac {1}{p}\sum \limits _{s < |z| \leqq r} {\nu |\xi (z){z^{ - p}}} } \right | \leqq A\lambda (Br){r^{ - p}} + A\lambda (Bs){s^{ - p}},} \\ \end {array}\] for all $r \geqq s > R$, all unit vectors $\xi$ in ${C^k}$, and all natural numbers p. Here $\nu |\xi$ represents the lifting of the divisor $\nu$ to the plane via the map $z \mapsto z\xi$ and ${N_{\nu |\xi }}$ is the valence function of that divisor. Analogous facts for functions of zero $\lambda$-type are also presented.