Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two
type: Item
instance of: scholarly article
title: Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two
zbMATH Open document ID: 0622.30028
DOI: 10.1007/BF02392256
published in: Acta Mathematica
publication date: 1987-01-01T00:00:00Z
review text: While developing ''Nevanlinna theory'' the Nevanlinna brothers formulated many conjectures. All of these were settled by now except for F. Nevanlinna's conjecture that a meromorphic function f(z) of finite order \(\lambda\) and with \(\sum_{a\in {\hat {\mathbb{C}}}}\delta (a,f)=2\) must satisfy \[ (1)\quad^ 2\lambda \text{ is an integer \(\geq 2\) and (2) }\delta (a,f)=p(a)/\lambda,\quad p(a)\in {\mathbb{Z}}, \] and all deficient values are asymptotic. The paper under review shows that this conjecture is correct. The principal tool is the theory of quasiconformal modifications of meromorphic functions developed by the author and A. Weitsman [see \textit{D. Drasin}, Acta Math. 138, 83-151 (1977; Zbl 0355.30028)]. The proof requires many delicate, intricately linked arguments leading up to the construction of a quasiconformal modification of \(f(z^ 2)\) which behaves essentially like \(\exp (z^{2\lambda})\) and from whose properties the truth of the conjecture follows.
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 4008765
Wikidata QID: Q56807198
author: David Drasin
reviewed by: W. H. J. Fuchs
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4008765