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On Mean Value Limits for Bounded Regular Functions

Published online by Cambridge University Press:  20 January 2009

N. A. Bowen
N. A. Bowen
Affiliation:
The University, Leicester
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The object of this note is to state and prove two theorems of the nature of Montel's Limit Theorem for a function which is regular and bounded in a region G, but involving as hypothesis the limit of a mean value of the function instead of the limit of the function itself. Theorem 2 below (with b = b′ = 1), in which G is a half-strip, was stated several years ago in a letter to A. J. Macintyre and myself from J. M. Whittaker, who added that it could be proved by integrating the inequality in Lemma 3 below (which is due to Dr Whittaker). As far as I can discover, such a proof still has not appeared in print; hence the present one.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 14 , Issue 2 , December 1964 , pp. 103 - 108
Copyright
Copyright © Edinburgh Mathematical Society 1964

References

REFERENCES

(1)Hall, T., Arkiv för Mat., Ast. och Fysik, 25A, No. 28, (1937), 1–8.Google Scholar
(2)Whittaker, J. M., Interpolatory Function Theory (Cambridge, 1935).Google Scholar
(3)Whittaker, J. M., Proc. London Math. Soc., (2) 41 (1936), 544–552.CrossRefGoogle Scholar