Koebe arcs and quasiregular mappings
According to a well known theorem of E. Lindelöf a bounded analytic function of the half-plane \(R^ 2_+=\{x\in R^ 2: x_ 2>0\}\) having a limit \(\alpha\) along a curve \(\gamma\) terminating at 0 has in fact an angular limit \(\alpha\) at 0. \textit{S. Rickman} [Ann. Acad. Sci. Fenn., Ser. A I 5, 185-196 (1980; Zbl 0453.30003)] has shown by an example that a similar result fails to hold for bounded quasiregular (qr) mappings in \(R^ n\) if \(n\geq 3\). In other words, for \(n\geq 3\) there exists a bounded qr mapping \(f:R^ n_+\to R^ n\) having a limit \(\alpha\) along a tangential curve \(\gamma\) at 0 but failing to have an angular limit \(\alpha\) at 0. As shown by the author in Trans. Am. Math. Soc. 263, 343- 354 (1981; Zbl 0461.30015) the situation is different for non-tangential curves [for a survey of this topic see the author, Complex analysis, Banach Cent. Publ. 11, 353-362 (1983; Zbl 0543.30015)]. In the paper under review the author proves that a bounded qr mapping of \(R^ n_+\), \(n\geq 2\), having a limit \(\alpha\) along a curve \(\gamma\) terminating at 0 has an angular limit \(\alpha\) at 0 as well provided that f(x) approaches \(\alpha\) fast enough along \(\gamma\). This result holds no matter whether \(\gamma\) is tangential or not while the required rate of convergence depends on the rate of tangentiality of \(\gamma\). In addition, it is shown that \(f\equiv\alpha \) if a somewhat faster rate of convergence is required. This last result is closely related to some well known theorems about bounded analytic functions due to M. Heins (1947), A. L. Shaginyan (1959), D. C. Rung (1968), and J. S. Hwang, F. Schnitzer and W. Seidel (1971).
scientific article; zbMATH DE number 3869685
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