Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two (Q1090803)

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.