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Mathematische Annalen

, Volume 111, Issue 1, pp 98–118 | Cite as

Some inequalities in the theory of functions

  • M. L. Cartwright
Article

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Literatur

  1. 1).
    P. Koebe,Gött. Nachrichten 1909, 68–76 (73).Google Scholar
  2. 2).
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    H. Frazer,Proc. London Math. Soc. (2)33 (1932) 77–81. [Added in the proofs.] Frazer has since shown, by methods which are elementary compared with those used here, that, ifw (z) does not take any value more thanp times, and does not take the value 0 more thanq times,q≦p, then (1.16) holds withC <Kμq, where µα = max(∣α0∣, ∣α1∣, ..., ∣αq∣. This result is not included in theorem I as it stands, except whenq=p. But, if we suppose, in addition, thatw(z) does not take the value 0 more thanq times, whereq<p, we may replace μ by μq. For in lemma 5 the order of the quasi-normal family is then less than or equal toq, and the result follows by the same method.Google Scholar
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    See P. Montel,Leçons sur les familles normales (Borel Series) (1927). S. 66–70.Google Scholar
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    Ther o in the statement of the theorem is equal to 1−(1−r o) cosα.Google Scholar

Copyright information

© Springer-Verlag 1935

Authors and Affiliations

  • M. L. Cartwright
    • 1
  1. 1.Cambridge(England)

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