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An analogue of the defect relation for the uniform metric. (English) Zbl 0905.30025
Let \(f\) be a meromorphic function. We use the following denotations: \[ \begin{aligned} M(r,\infty,f) & =\sup_\theta \bigl| f(re^{i\theta}) \bigr|,\;M(r,a,f)= M\left(r,\infty, {1\over f-a} \right),\\ A(r,f) & ={1\over\pi} \iint_{| z|\leq r} {\bigl| f'(z)\bigr |^2 \over \biggl(1+ \bigl| f(z)\bigr|^2 \biggr)^2} dxdy,\;z=x+iy,\\ b(a,f) & =\varliminf_{r\to\infty} {\ln^+ M(r,a,f) \over A(r,f)}. \end{aligned} \] The author obtains the inequality \(\sum_a b(a,f)\leq 2\pi\) if for every \(a\in \overline \mathbb{C}\), \(b(a,f)\leq 2\pi\). Thus, we have an analogue of the Nevanlinna defect relation for the values \(b(a,f)\). The necessity of introducing \(b(a,f)\) appears evident after the paper of W. Bergweiler and H. Bock (1994).

30D30 Meromorphic functions of one complex variable, general theory
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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