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On Montel’s Theorem

Published online by Cambridge University Press:  22 January 2016

Yoshiro Kawakami
Yoshiro Kawakami*
Affiliation:
Seikei University, Tokyo
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In this note we shall prove a theorem which is related to Montel’s theorem [1] on bounded regular functions. Let E be a measurable set on the positive y-axis in the z( = x + iy)-plane, E(a, b) be its part contained in 0 ≦ a ≦ y ≦ b, and ∣E(a, b)∣ be its measure. We define the lower density of E at y = 0 by

LEMMA, Let E be a set of positive lower density λ at y = 0. Then E contains a subset E1 of the same lower density at y = 0 such that E1 ∪ {0} is a closed set.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 10 , June 1956 , pp. 125 - 127
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1956

References

[ 1 ] Montel, P., Sur les familles de fonctions analytiques qui admettent des valeurs exceptionelles dans un domaine, Ann. Sci. Ecole Norm. Sup. (3), 23 (1912), pp. 487–535.CrossRefGoogle Scholar