Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-02-05T14:42:47.470Z Has data issue: false hasContentIssue false
  • English
  • Français

Complex-Harmonic Meier’s Theorem

Published online by Cambridge University Press:  22 January 2016

Shinji Yamashita
Shinji Yamashita*
Affiliation:
Mathematical Institute, Tôhoku University, Sendai 980, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1. Fatou’s theorem is true for a bounded complex-valued harmonic function in the disk D: |z|<1. One asks naturally: “Is Meier’s topological analogue of Fatou’s theorem (simply, “MF theorem”; [14, p. 330, Theorem 6], cf. [10, p. 154, Theorem 8.9]) true for a bounded complex-valued harmonic function in D?” We shall give the affirmative answer to this question. Furthermore, the horocyclic MF theorem [2, p. 14, Theorem 5] in the complex-harmonic form will be proved in parallel.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 43 , September 1971 , pp. 161 - 165
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Bagemihl, F., Meier points of holomorphic functions, Math. Ann. 155 (1964), 422–424.Google Scholar
[2] Bagemihl, F., Horocyclic boundary properties of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A-I. Math. No. 385 (1966), 18 pp.Google Scholar
[3] Bagemihl, F., Chordal limits of holomorphic functions at Plessner points, J. Sci. Hiroshima Univ. Ser. A-I. 30 (1966), 109–115.Google Scholar
[4] Bagemihl, F., On the sharpness of Meier’s analogue of Fatou’s theorem, Israel J. Math. 4 (1966), 230–232.CrossRefGoogle Scholar
[5] Bagemihl, F., Generalizations of two theorems of Meier concerning boundary behavior of meromorphic functions, Publ. Math. Debrecen, 14 (1967), 53–55.Google Scholar
[6] Bagemihl, F., An improvement on Plessner’s theorem, Math. Z. 110 (1969), 177–179.Google Scholar
[7] Bagemihl, F., Meier points and horocyclic Meier points of continuous functions, Ann. Acad. Sci. Fenn. Ser. A-I. Math. No. 461 (1970), 7 pp.Google Scholar
[8] Collingwood, E.F., Cluster sets of arbitrary functions, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 1236–1242.CrossRefGoogle ScholarPubMed
[9] Collingwood, E.F., Cluster set theorems for arbitrary functions with applications to function theory, Ann. Acad. Sci. Fenn. Ser. A-I. Math. No. 336/8 (1963), 15 pp.Google Scholar
[10] Collingwood, E.F. and Lohwater, A.J., The theory of cluster sets, Cambridge (1966).Google Scholar
[11] Dragosh, S., Horocyclic boundary behavior of meromorphic functions, J. Analyse Math. 22 (1969), 37–48.Google Scholar
[12] Dragosh, S., Horocyclic cluster sets of functions defined in the unit disc, Nagoya Math. J. 35 (1969), 53–82.Google Scholar
[13] Heinz, E., On one-to-one harmonic mappings, Pacific J. Math. 9 (1959), 101–105.Google Scholar
[14] Meier, K.E., Ueber die Randwerte der meromorphen Funktionen, Math. Ann. 142 (1961), 328–344.Google Scholar
[15] Yamashita, S., Quasi-conformal extension of Meier’s theorem, Proc. Japan Acad. 46 (1970), 323–325.Google Scholar
[16] Yamashita, S., Cluster sets of algebroid functions, Tôhoku Math. J. 22 (1970), 273–289.Google Scholar
[17] Yamashita, S., Boundary behaviour of functions harmonic in the unit ball, Proc. Japan Acad. 46 (1970), No. 6, 490–493.Google Scholar
[18] Yamashita, S., On Fatou- and Plessner-type theorems, Proc. Japan Acad. 46 (1970), No. 6, 494–495.Google Scholar