Asymptotic values and Baire category
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- by Chaim Mida
- Proc. Amer. Math. Soc. 41 (1973), 492-494
- DOI: https://doi.org/10.1090/S0002-9939-1973-0324046-0
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Abstract:
Let $f$ be meromorphic in the unit disc, and let $\alpha$ be a complex number. Given $\varepsilon > 0$, let ${T_\varepsilon }(\alpha )$ denote the set of points ${e^{i\theta }}$ for which the cluster set ${C_\mathcal {L}}(f,{e^{i\theta }})$ lies in the $\varepsilon$-neighbourhood of $\alpha$ for some arc $\mathcal {L} \to {e^{i\theta }}$. Then a sufficient condition that the set of points on the unit circle at which $f$ possesses point-asymptotic value $\alpha$ be of first category is that ${T_\varepsilon }(\alpha )$ contains no arc for some $\varepsilon > 0$.References
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- A. J. Lohwater, Some function-theoretic results involving Baire category, Topics in analysis (Colloq. Math. Anal., Jyväskylä, 1970) Lecture Notes in Math., Vol. 419, Springer, Berlin, 1974, pp. 253–259. MR 0377060
- E. C. Titchmarsh, Han-shu lun, Science Press, Peking, 1964 (Chinese). Translated from the English by Wu Chin. MR 0197687
- Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 492-494
- MSC: Primary 30A72
- DOI: https://doi.org/10.1090/S0002-9939-1973-0324046-0
- MathSciNet review: 0324046