## The multiplicative Cousin problem and a zero set for the Nevanlinna class in the polydisc

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- by Sergio E. Zarantonello PDF
- Trans. Amer. Math. Soc.
**200**(1974), 291-313 Request permission

## Abstract:

Let $\Omega$ be a polydomain in ${{\mathbf {C}}^n}$, the Nevanlinna class $N(\Omega )$ consists of all holomorphic functions $f$ in $\Omega$ such that ${\log ^ + }|f|$ has an $n$-harmonic majorant in $\Omega$. Let ${U^n}$ be the open unit polydisc $\{ z \in {{\mathbf {C}}^n}:|{z_1}| < 1, \cdots ,|{z_n}| < 1\}$. THEOREM 1.*Given an open covering ${({\Omega _\alpha })_{\alpha \in A}}$ of the closure ${\bar U^n}$ of the polydisc, consisting of polydomains, and for each $\alpha \in A$ a function ${f_\alpha } \in N({\Omega _\alpha } \cap {U^n})$ such that for all $\alpha ,\beta \in A,{f_\alpha }f_\beta ^{ - 1}$ is an invertible element of $N({\Omega _\alpha } \cap {\Omega _\beta } \cap {U^n})$. There exists a function $F \in N({U^n})$ such that for all $\alpha \in A,Ff_\alpha ^{ - 1}$ is an invertible element of $N({\Omega _\alpha } \cap {U^n})$*. This result enables us to find the following sufficient condition for the zero sets of $N({U^n})$: THEOREM 2.

*Let $f$ be a holomorphic function in ${U^n},n \geqslant 2$. If there exists a constant $0 < r < 1$ and a continuous function $n:[r,1) \to [r,1)$ such that*\[ |{z_n}| \leqslant n\left ( {\frac {{|{z_1}| + \cdots + |{z_{n - 1}}|}}{{n - 1}}} \right )\]

*for all points $({z_1}, \cdots ,{z_n})$ satisfying $|{z_1}| > r, \cdots ,|{z_n}| > r$ and $f({z_1}, \cdots ,{z_n}) = 0$, then $f$ has the same zeros as some function $F \in N({U^n})$. In the above if ${\overline {\lim } _{\lambda \to 1}}n(x) < 1$, then $Z(f)$ is a Rudin variety in which case there is a bounded holomorphic function with the same zeros as $f$.*

## References

- Lars V. Ahlfors,
*Complex analysis: An introduction of the theory of analytic functions of one complex variable*, 2nd ed., McGraw-Hill Book Co., New York-Toronto-London, 1966. MR**0188405** - P. S. Chee,
*On the generalized Blaschke condition*, Trans. Amer. Math. Soc.**152**(1970), 227–231. MR**268405**, DOI 10.1090/S0002-9947-1970-0268405-5 - Pak Soong Chee,
*The Blaschke condition for bounded holomorphic functions*, Trans. Amer. Math. Soc.**148**(1970), 249–263. MR**262541**, DOI 10.1090/S0002-9947-1970-0262541-5 - Walter Rudin,
*Function theory in polydiscs*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0255841** - Walter Rudin,
*Zero-sets in polydiscs*, Bull. Amer. Math. Soc.**73**(1967), 580–583. MR**210934**, DOI 10.1090/S0002-9904-1967-11758-0 - E. L. Stout,
*The second Cousin problem with bounded data*, Pacific J. Math.**26**(1968), 379–387. MR**235155**
S. E. Zarantonello,

*Sobre los ceros de funciones analiticas acotadas de varias variables complejas*, An. Inst. Mat. Univ. Nac. Autónoma México (to appear).

## Additional Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**200**(1974), 291-313 - MSC: Primary 32A30
- DOI: https://doi.org/10.1090/S0002-9947-1974-0355092-4
- MathSciNet review: 0355092