Functions meromorphic in the unit circle

  • Kiyoshi Noshiro
Part of the Ergebnisse der Mathematik und Ihrer Grenzgebiete book series (MATHE2, volume 28)

Abstract

Let w = f(z) be a bounded regular function in the unit circle |z| < 1.

Keywords

Lime Cose Larg sinO 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. 1.
    Nevanlinna [1] was the first to point out the interest which lies in the class (U). Cf. Hössjer and Frostman [1], Seidel [2], Frostman [1], Noshiro [3, 11], Calderón, Gonzales Domingues and Zygmund [1], Lohwater [2, 3, 7], Ohtsuka [9].Google Scholar
  2. 2.
    F. Riesz [1].Google Scholar
  3. 3.
    Herglotz [1].Google Scholar
  4. 1.
    Schlesinger and Plessner [1], § 43.Google Scholar
  5. 2.
    Noshiro [11]. Cf. also Lohwater [2].Google Scholar
  6. 3.
    Hössjer [1], Kawakami [1], Kametani-Ugaeri [1], Lohwater-Seidel [1], Ohtsuka [2], Tsuji [17].Google Scholar
  7. 2.
    Seidel [2].Google Scholar
  8. 1.
    This proof is due to Bagemihl[1].Google Scholar
  9. 1.
    Cf. Theorem 4, § 4, II and Theorem 7, § 4, II.Google Scholar
  10. 1.
    It is easy to see that Theorem 7 remains valid in the case of a function of generalized class (U). Google Scholar
  11. 2.
    Cf. Theorem 1, § 3, II and Theorem 4, § 4, II.Google Scholar
  12. 1.
    As for related results, cf. Ohtsuka [9, 15], Lohwater [3–8], Storvick [2], Lehto [7, 9].Google Scholar
  13. 1.
    Cf. Lehto [8] on the construction of such functions.Google Scholar
  14. 1.
    Collingwood-Cartwright [1].Google Scholar
  15. 1.
    For a related theorem, see Theorem 14, §4, III (Bagemihl-Seidel [3]). Cf. also Heins [4] in the case of an entire function.Google Scholar
  16. 1.
    Cf., for example, Collingwood and Cartwright [1], p. 109 and Noshiro [2], p. 230.Google Scholar
  17. 1.
    This theorem is closely related to results of Seidel [2], Ohtsuka [1, 9, 15], Lohwater [2, 7, 8], Lehto [3, 7], Noshiro [11].Google Scholar
  18. 1.
    See Paragraph 3, § 2, I. Cf. Collingwood-Cartwright [1], pp. 96–98.Google Scholar
  19. 1.
    The following is similar to and much easier than the arguments used in the proof of Theorem 10, § 1.Google Scholar
  20. 1.
    Lusin and Privaloff [1].Google Scholar
  21. 2.
    Valiron [1], Bagemihl-Erdös-Seidel [1]; for the related results, see §4.Google Scholar
  22. 3.
    Bagemihl and Seidel [1], p. 1072.Google Scholar
  23. 1.
    See the reconstructed proof in Collingwood [10].Google Scholar
  24. 1.
    This method has been used by Collingwood to prove Lemma 2, § 3.Google Scholar
  25. 1.
    This means the special case where E consists of a single point, of Theorem 1, §3,11.Google Scholar
  26. 2.
    Cf. Theorem 8, § 1.Google Scholar
  27. 1.
    See the detail of the proof in Bagemihl-Seidel [8]. For related theorems, cf. Bagemihl-Erdös-Seidel [1].Google Scholar
  28. 1.
    See, e. g., Kuratowski [1], p. 117.Google Scholar
  29. 1.
    In case every K n is a single point, this result has been obtained in Bagemihl-Seidel [2].Google Scholar
  30. 2.
    In the proof, the Hahn-Mazurkiewicz theorem (cf. C. Kuratowski [2], p. 188) is used.Google Scholar
  31. 1.
    For the extension of this theorem to the case of a multiply connected domain, see Bagemihl-Seidel [5].Google Scholar
  32. 2.
    For the extension of this theorem, see Bagemihl-Seidel [4].Google Scholar
  33. 1.
    As to its related results and extensions, see Jenkins [1], Piranian and Rudin [1], Piranian and Shields [1], Bagemihl [4], Piranian [1].Google Scholar
  34. 2.
    As to its related results and extensions, see Herzog and Piranian [1], Rudin [4], Bagemihl [5].Google Scholar
  35. 1.
    As is well-known, Mme. Schwartz, Teichmüller and Hayman have given counterexamples of meromorphic functions for which an exceptional value (in the sense of Nevanlinna) is not asymptotic (cf. Wittich [l]).Google Scholar
  36. 2.
    Letho [7], Theorem 9, p. 41. Huckemann [1] has shown that indirect critical singularities may give rise to deficient values.Google Scholar
  37. 3.
    In the case of the unit disc, the writer said such a function to belong to class (A) and obtained some results (see, Noshiro [3]).Google Scholar
  38. 4.
    For the proof, see Lehto-Virtanen [1], Theorem 1, pp. 49–52.Google Scholar
  39. 5.
    An entirely similar argument was used in Noshiro [3]. See also recent related results of Seidel [3].Google Scholar
  40. 1.
    Ahlfors [7], p. 169. Cf. also Marty [1].Google Scholar
  41. 2.
    Ahlfors [2].Google Scholar

Copyright information

© Springer-Verlag OHG. Berlin · Göttingen · Heidelberg 1960

Authors and Affiliations

  • Kiyoshi Noshiro

There are no affiliations available

Personalised recommendations