Nevanlinna functions as quotients

Doubtsov, Evgueni

doi:10.1090/S0002-9939-00-05446-0

Nevanlinna functions as quotients
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Proc. Amer. Math. Soc. 128 (2000), 2899-2901 Request permission

Abstract:

Let $f$ be a holomorphic function in the unit ball. Then $f$ is a Nevanlinna function if and only if there exist Smirnov functions $f_+$, $f_-$ such that $f = f_+/f_-$ and $f_-$ has no zeros in the ball.

References

A. G. Vitushkin (ed.), Several complex variables. II, Encyclopaedia of Mathematical Sciences, vol. 8, Springer-Verlag, Berlin, 1994. Function theory in classical domains. Complex potential theory; A translation of Current problems in mathematics. Fundamental directions, Vol. 8 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985 [ MR0850484 (87f:32004)]; Translation by P. M. Gauthier and J. R. King; Translation edited by G. M. Khenkin and A. G. Vitushkin. MR 1298905, DOI 10.1007/978-3-642-57882-3
E. Doubtsov, Henkin measures, Riesz products and singular sets, Ann. Inst. Fourier (Grenoble) 48 (1998), 699–728.
M. Seetharama Gowda, Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of $\textbf {C}^{n}$ $(n>1)$, Trans. Amer. Math. Soc. 277 (1983), no. 1, 203–212. MR 690048, DOI 10.1090/S0002-9947-1983-0690048-9
G. M. Henkin, H. Lewy’s equation, and analysis on a pseudoconvex manifold. II, Mat. Sb. (N.S.) 102(144) (1977), no. 1, 71–108, 151 (Russian). MR 0473232
Walter Rudin, Zeros of holomorphic functions in balls, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math. 38 (1976), no. 1, 57–65. MR 0393538
Henri Skoda, Valeurs au bord pour les solutions de l’opérateur $d''$, et caractérisation des zéros des fonctions de la classe de Nevanlinna, Bull. Soc. Math. France 104 (1976), no. 3, 225–299 (French). MR 450620