Advertisement

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

Unit Circle Meromorphic Function Blaschke Product Radial Limit Capacity Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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