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On positive harmonic functions

Published online by Cambridge University Press:  24 October 2008

A. C. Allen
A. C. Allen
Affiliation:
PeterHouseCambridge
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Extract

Any harmonic function which is defined and positive in the half-plane η > 0 may be expressed by

where C is a non-negative number, and G(x) is a bounded non-decreasing function. For a simple proof see Loomis and Widder (2). Let us write

where w(z) is a regular function of z in η > 0, and satisfies the following conditions: (i) w(z) is real and continuous at all points of the open interval (a, b) of the real axis [the interval may be unbounded]; (ii) there exists a simply connected domain Δ, lying in the half-plane η > 0, whose frontier contains the interval (a, b) of the real axis and which is mapped ‘simply’ on the half-plane υ > 0 by the conformal transformation w = w(z).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 4 , October 1952 , pp. 571 - 577
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Loomis, L. H.Trans. Amer. math. Soc. 53 (1943), 239–50.Google Scholar
(2)Loomis, L. H. and Widder, D. V.Duke math. J. 9 (1942), 643–5.Google Scholar
(3)Verblunsky, S.Proc. Camb. phil. Soc. 44 (1948), 289–91.CrossRefGoogle Scholar