Hayashi, Mikihiro; Nakai, Mitsuru A uniqueness theorem and the Myberg phenomenon. (English) Zbl 0977.30032 J. Anal. Math. 76, 109-136 (1998). The authors continue their studies on bounded holomorphic functions on Riemann surfaces [see for example Pac. J. Math 134, 261-73 (1988; Zbl 0627.30031); Pitman Res. Notes Math. Sci. 212, 1-12 (1989; Zbl 0698.30036); Trans. Am. Math. Soc. 333, No. 2, 799-819 (1992; Zbl 0759.30018)]. Let \(H^{\infty}(W)\) be the space of bounded holomorphic functions on a Riemann surface \(W\) and \(\widetilde W\) an unlimited covering surface of \(W\) with projection map \(\varphi\). The Myrberg phenomenon is said to occur if \(H^{\infty}(\widetilde W)=H^{\infty}(W) \circ \varphi\). A Zalcman domain \(R\) is a subdomain of the punctured unit disc obtained by deleting a sequence of disjoint discs converging to \(0\) with centers on the positive real axis. Let \((\widetilde R , R, \varphi)\) be the unlimited two-sheeted covering surface of \(R\) with projection map \(\varphi\). In previous papers, the authors proves that the irregularity of \(z=0\) for the region \(R\) (in the sense of potential theory) is sufficient for the Myrberg phenomenon to occur for \((\widetilde R , R, \varphi)\). The main purpose of the present paper is to prove that this sufficient condition is not necessary. The well written paper contains numerous ideas for further development of the theory. Reviewer: H.Köditz (Hannover) Cited in 2 Documents MSC: 30H05 Spaces of bounded analytic functions of one complex variable 30D50 Blaschke products, etc. (MSC2000) 30F99 Riemann surfaces 46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces Keywords:bounded analytic functions on Riemann surfaces Citations:Zbl 0627.30031; Zbl 0759.30018; Zbl 0698.30036 PDFBibTeX XMLCite \textit{M. Hayashi} and \textit{M. Nakai}, J. Anal. Math. 76, 109--136 (1998; Zbl 0977.30032) Full Text: DOI References: [1] Carleman, T., Les fonctions quasi-analytiques (1926), Paris: Gauthier-Villars, Paris · JFM 52.0255.02 [2] Fisher, S., Function Theory on Planar Domains (1983), New York: Wiley, New York · Zbl 0511.30022 [3] Hayashi, M., Smoothness of analytic functions at boundary points, Pacific J. Math., 67, 171-202 (1976) · Zbl 0348.30034 [4] M. Hayashi and T. Kato,Point separation of a two-sheeted disc by bounded analytic functions, Hokkaido Math. J. (1998) (to appear). · Zbl 0920.30025 [5] Hayashi, M.; Nakai, M., Point separation by bounded analytic functions of a covering Riemann surface, Pacific J. Math., 134, 261-273 (1988) · Zbl 0627.30031 [6] M. Hayashi and M. Nakai,On the Myrberg type phenomenon, inAnalytic Function Theory of One Complex Variable (Y. Komatsu, K. Niino and C. C. Yang, eds.), Pitman Research Notes in Math. Sci.212 (1989), 1-12. · Zbl 0698.30036 [7] Hayashi, M.; Nakai, M.; Segawa, S., Bounded analytic functions on two sheeted discs, Trans. Amer. Math. Soc., 333, 799-819 (1992) · Zbl 0759.30018 [8] Hayashi, M.; Nakai, M.; Segawa, S., Two-sheeted discs and bounded analytic functions, J. Analyse Math., 61, 293-325 (1993) · Zbl 0795.30029 [9] Landkof, N. S., Foundations of Modern Potential Theory (1972), Berlin: Springer, Berlin · Zbl 0253.31001 [10] Myrberg, P. J., über die Analytische Fortsetzung von beschrÄnkten Funktionen, Ann. Acad. Sci. Fenn. Ser. A, I Math., 58, 1-7 (1949) · Zbl 0034.05203 [11] Nakai, M., Valuations on meromorphic functions of bounded type, Trans. Amer. Math. Soc., 309, 231-252 (1988) · Zbl 0656.30026 [12] Nakai, M., On the existence of H^∞-barrier, Rev. Roumanie Math. Pures Appl., 36, 161-167 (1991) · Zbl 0752.30022 [13] Ostrowski, A., über quasianalytische Funktionen und Bestimmtheit asymptotischer Entwickelungen, Acta Math., 53, 181-266 (1929) · JFM 55.0184.04 [14] Sario, L.; Nakai, M., Classification Theory of Riemann Surfaces (1970), Berlin: Springer, Berlin · Zbl 0199.40603 [15] Tsuji, M., Potential Theory in Modern Function Theory (1975), New York: Chelsea, New York · Zbl 0322.30001 [16] Zalcman, L., Bounded analytic functions on domains of infinite connectivity, Trans. Amer. Math. Soc., 144, 241-269 (1969) · Zbl 0188.45002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.