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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

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