Vol. 27, No. 3, 1968

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On a paper of Rao

Myron Goldstein

Vol. 27 (1968), No. 3, 497–500

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.

Mathematical Subject Classification
Primary: 30.45
Received: 17 January 1968
Published: 1 December 1968
Myron Goldstein