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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.
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