Vol. 15, No. 4, 1965

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ISSN: 0030-8730
Φ-bounded harmonic functions and classification of Riemann surfaces

Mitsuru Nakai

Vol. 15 (1965), No. 4, 1329–1335

Let Φ(t) be a nonnegative real valued function defined for t in [0,) such that Φ(t) is unbounded in [0,) and bounded in a neighborhood of a point in [0,). A harmonic function u on a Riemann surface R is said to be Φ-bounded if the composite function Φ(|u|) has a harmonic majorant on R. Denote by OHΦ the class of all Riemann surfaces on which every Φ-bounded harmonic function reduces to a constant. The main result in this paper is the following: OHΦ = OHP (resp. OHB) if and only if d(Φ) < (resp. d(Φ) = ), where d(Φ) = limsupt→∞Φ(t)∕t. This is the best possible improvement of a result of M. Parreau.

We also prove a similar theorem for the classification of subsurfaces of Riemann surfaces using Φ-bounded harmonic functions vanishing on the relative boundaries of subsurfaces.

The chief tool of our proof is the theory of Wiener compactifications of Riemann surfaces.

Mathematical Subject Classification
Primary: 30.45
Received: 5 September 1964
Published: 1 December 1965
Mitsuru Nakai