Meromorphic Functions on Certain Riemann Surfaces

1. Throughout the paper we shall denote by R a Riemann surface. For a domain Í2 in P, we represent by AB(Q) the class of all the singlevalued bounded analytic functions on the closure Ü. For a meromorphic function / on a domain ß, we use the notation viw\f, Q.) to express the number of times that/ attains w in ß. Definition 1. We say that REWIb if the maximum principle suplen \fip)\ =sup3,ean \fip)\ holds for every/ in the class AB(Q) for every fi ER with compact relative boundary 50. Definition 2. We say that P£2IB if, for every flCP with compact dû, every function in the class AB(Í2) has its limit at every ideal boundary point in the sense of Kerékjártó-Stoilow. Definition 3. We say that RE&b if, for every fí CP with compact dû, the cluster set P/(ß) of every/ in the class AB(fl) attached to the ideal boundary is a totally disconnected set in the complex w-plane. From the definitions, we have immediately, ©bCSIb^Oab, ©b^Oab and SWb ÍOabIf R has only a countable number of ideal boundary points, we see easily that RE'S.b implies P£jDb. It is also evident that, if the surfaces are restricted to plane domains, then 50Îb = OabC©bC2Ib.