Mathematische Annalen

, Volume 111, Issue 1, pp 98–118

# Some inequalities in the theory of functions

• M. L. Cartwright
Article

## Literatur

1. 1).
P. Koebe,Gött. Nachrichten 1909, 68–76 (73).Google Scholar
2. 2).
L. Bieberbach,Lehrbuch der Funktionentheorie, Vol.II. (Berlin, 1927), 82–92.Google Scholar
3. 3).
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5. 5).
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6. 6).
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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8. 8).
See lemma 5 and footnote.Google Scholar
9. 9).
See E. C. Titchmarsh,Theory of Functions, Oxford (1932), 121, 122.Google Scholar
10. 10).
L. Ahlfors,Acta Soc. Sci. Fen., Nova Series A,1. The calculations are practically the same as in Ahlfors' work, I include them merely for completeness.Google Scholar
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13. 13).
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14. 14).
loc. cit. P. Montel,Leçons sur les familles normales (Borel Series) (1927). S. 22.Google Scholar
15. 15).
See M. L. Cartwright,Quarterly Journal (Oxford Series)1 (1930), S. 44.Google Scholar
16. 16).
Ther o in the statement of the theorem is equal to 1−(1−r o) cosα.Google Scholar