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Some Generalizations of Montel's Theorem

Published online by Cambridge University Press:  24 October 2008

Mary L. Cartwright
Mary L. Cartwright
Affiliation:
Girton College
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Extract

1. There are two well-known theorems on the limit of a bounded function at a point.

Montel's Theorem. Suppose that f (z) is regular for | arg z | ≤ α, | z | ≤ 1, except perhaps at z = 0, and that f(z) is bounded in that region. Suppose also that f(z) → l as z → 0 along arg z = β, where | β | < α. Then f(z) → l as z → 0 uniformly for | arg z | ≤ α − δ for every δ > 0.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 1 , January 1935 , pp. 26 - 30
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* See Dienes, P., The Taylor Series, Oxford (1931), 456.Google Scholar

† Loc. cit. p. 46, see also Julia, G., Leçons sur les fonctions uniformes, Borel Series (1924), 94–5.Google Scholar

‡ Hardy, G. H. and Littlewood, J. E., Proc. London Math. Soc. (2), 18 (1918), 205–35Google Scholar lemmas ε and δ.

* Pólya, G., Math. Zeitschrift, 29 (1929), 549–640 (633)CrossRefGoogle Scholar. See also Landau, E., Göttinger Nachrichten (1930), 1–9Google Scholar. The best possible value of K has been obtained by Nevanlinna, R., Göttinger Nachrichten (1933), 103–15.Google Scholar

† Milloux, H., Journal de Math. (9), 3 (1924), 345–401.Google Scholar

‡ Landau, E., Göttinger Nachrichten (1930), 1–9.Google Scholar

§ Milloux, H., Matematica, 4 (1930), 182–5.Google Scholar

* See Montel, P., Leçons sur les families normales, Borel Series (1927), 193.Google Scholar