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In this paper, we give an
internal proof of Rao’s theorem on meromorphic functions of bounded characteristic,
i.e., a proof not using uniformization.
In addition, we discuss the classification theory of Riemann surfaces as it pertains
to the class OL of hyperbolic Riemann surfaces which admit no nonconstant
Lindelöfian meromorphic functions. In particular, we show that UHB ⊂ OL where
UHB denotes the class of hyperbolic Riemann surfaces on which there exist at least
one bounded MHB minimal function.
We also show that there is no inclusion relation between OL and OHDn, n a
natural number, where OHDn denotes the class of hyperbolic Riemann surfaces for
which the dimension of the vector lattice HD is at most n.
Finally, we generalize the F. and M Riesz theorem for H1 of the unit disc to
arbitrary open hyperbolic Riemann surfaces.
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