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Published online by Cambridge University Press: 24 October 2008
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
* 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.
