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On the magnitudes of deviations of meromorphic functions. (Russian) Zbl 0697.30033

Interesting results concerning defects in the Petrenko sense are proved in this paper. Among these results we mention the following ones.
1. Let f(z) be a meromorphic function in the complex plane, \(\lambda\) be its lower order, \[ \beta (\infty,f)=\liminf_{r\to \infty}(\max \{\log^+| f(z)|:\quad | z| =r\})/(T(r,f)), \] \(\beta\) (a,f)\(=\beta (\infty,(f-a)^{-1})\), \(a\in {\mathbb{C}}.\)
Then \[ \sum_{a\in {\bar {\mathbb{C}}}}\beta (a,f)\leq 2\pi \lambda,\quad \lambda <1/2,\quad \sum_{a\in {\bar {\mathbb{C}}}}\beta (a,f)\leq (2\pi \lambda)/(\sin \pi \lambda),\quad \lambda \geq 1/2. \] 2. Define in the unit disk for the meromorphic function f(z) with unbounded characteristic T(r,f) \[ {\hat \beta}(\infty,f)=\liminf_{r\to \infty}((1-r)\max \{\log^+| f(z)|:\quad | z| =r\})/(T(r,f)). \] Then \({\hat \beta}\)(\(\infty,f)\leq \pi \lambda \cos^{-1-\lambda}(\pi /2(\lambda +1))\). The proofs use the technique of the *-function.
Reviewer: M.Sodin

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Keywords:

*-function
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