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Value distribution theory for meromorphic functions of slow growth in the disk. (English) Zbl 0613.30032
Let \(\rho\) (\(\rho\) ’) be the order of f(z)(f’(z)). If f is meromorphic in the plane, then \(\rho =\rho '\), but the relation is more complicated when f is meromorphic in \({\mathbb{D}}=\{| z| <1\}\). The authors set \[ \alpha =\alpha (f)=\limsup T(r,f)\{\log (1/(1-r))\}^{-1}\quad (r\to 1); \] when \(\alpha =\infty\), the analogy with functions meromorphic in the plane is clearest. Let \({\mathbb{F}}\) be those f(z) with \(\alpha <\infty\). Then \[ \alpha (f')\leq \alpha (f)\{1+k(f)\}+1, \] where \[ k(f)=\limsup \bar N(r,\infty)(T(r,f)+1)^{-1}\quad (r\to 1), \] and this estimate is sharp. This gives refinements of the deficiency relation for functions in \({\mathbb{F}}.\)
The necessary examples are gap series, or depend on a construction of W. K. Hayman [Acta Math. 112, 181-214 (1964; Zbl 0141.078)].
Reviewer: D.Drasin

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D40 Cluster sets, prime ends, boundary behavior
gap series