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On Positive Harmonic Functions in a Half-Plane

Published online by Cambridge University Press:  24 October 2008

S. Verblunsky
S. Verblunsky
Affiliation:
Magdalene College
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Extract

1. Let ξ, η denote the rectangular Cartesian coordinates of a point in a plane. Let J (ξ, η) denote a harmonic function which is positive in the half-plane η > 0. In this paper, we first show (Theorem I) that every such function J determines a non-negative number d, and a bounded non-diminishing function G(x), such that

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 4 , October 1935 , pp. 482 - 507
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* For further information concerning this case, see a paper by the author in Proc. London Math. Soc. (2), 37 (1934), 338–82Google Scholar, Theorem XXIV.

* Cf. Verblunsky, , Proc. London Math. Soc. (2), 38 (1934), 125–57Google Scholar, Theorem A.

* Hobson, , Theory of Functions of a Real Variable, 2, 246.Google Scholar

* Cf. Hardy, Littlewood and Pólya, , Inequalities (1934), pp. 196–8.Google Scholar

* Riesz, M.. Acta Math. 49 (1927), 465–97.CrossRefGoogle Scholar

* Cf. Zygmund, , Trigonometrical Series (1935), p. 48.Google Scholar

† Hardy, and Littlewood, , Journal London Math. Soc. 6 (1931), 281–6CrossRefGoogle Scholar, Theorem 1.

‡ Hardy, and Littlewood, , Journal London Math. Soc. 1 (1926), 134–8.CrossRefGoogle Scholar

* Hardy, Littlewood and Pólya, loc. cit. pp. 239, 246.

* Hardy, Littlewood and Pólya, loc. cit. 225, Theorem 346.