Hayashi, Mikihiro; Nakai, Mitsuru A principal function problem and the field of meromorphic functions. (English) Zbl 1294.30090 J. Anal. Math. 121, 127-140 (2013). Summary: Let \(R\) be an open Riemann surface and \(A\) an open subset of \(R\). For a given meromorphic function \(s\) on \(A\), we show, under an additional condition on \(A\), that there exists a meromorphic function \(q\) on \(R\) such that both \(q/s\) and \(s/q\) are bounded functions on \(A\). MSC: 30F99 Riemann surfaces 30D30 Meromorphic functions of one complex variable (general theory) Keywords:open Riemann surface; meromorphic function PDFBibTeX XMLCite \textit{M. Hayashi} and \textit{M. Nakai}, J. Anal. Math. 121, 127--140 (2013; Zbl 1294.30090) Full Text: DOI References: [1] H. Behnke and K. Stein, Entwicklung analytischer Funktionen auf Riemannschen Flächen, Math. Ann. 120 (1949), 430–461. · Zbl 0038.23502 [2] E. Bishop, Analyticity in certain Banach spaces, Trans. Amer. Math. Soc. 102 (1962), 507–544. · Zbl 0112.07301 [3] T. W. Gamelin, Uniform Algebras, Prentice-Hall, 1969. · Zbl 0213.40401 [4] T.W. Gamelin and M. Hayashi, The algebra of bounded analytic functions on a Riemann surface, J. Reine Angew. Math. 382 (1987), 49–73. · Zbl 0635.30039 [5] M. Hayashi, Hardy classes on Riemann surfaces, Thesis, UCLA, 1979. [6] M. Hayashi, The maximal ideal space of the bounded analytic functions on a Riemann surface, J. Math. Soc. Japan 39 (1987), 337–344. · Zbl 0598.30070 [7] M. Hayashi, Classification of Riemann surfaces with a nonconstant meromorphic function bounded at infinity, J. Contemp. Math. Anal. 32 (1997) no. 4, 37–44. · Zbl 0916.30025 [8] B. Rodin and L. Sario, Principal Functions, Van Nostrand, 1968. · Zbl 0159.10701 [9] H. L. Royden, Algebras of bounded analytic functions on Riemann surfaces, Acta Math. 114 (1965), 113–142. · Zbl 0146.30501 [10] L. Sario and M. Nakai, Classification Theory of Riemann Surfaces, Springer, 1970. · Zbl 0199.40603 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.