On asymptotic values of analytic functions on Riemann surfaces

Content Provider	Semantic Scholar
Author	Niimura, Mikio
Copyright Year	1976
Abstract	Some generalizations of Lindelof's theorems on asymptotic values of bounded analytic functions are given on subregions of Riemann surfaces. Let R be an open Riemann surface. Let R* denote a metrizable compactification of R, and put A = R*R. A means the closure of a set A C R* with respect to R *. aA means the relative boundary of A c R with respect to R. Let G be a region, which is not relatively compact on R, with the property that aG consists of a finite number of noncompact Jordan arcs C0 (n = 1, 2, . .. , N), and that G n A is a single point p. Each point q of aG is accessible in G. It is said that a Jordan arc J a = g(t) (O < t < 1) decides an accessible boundary point q(J) in G, when J c G and limt1g(t) = q. Let Jordan arcs J1 and J2 decide accessible boundary points q(J1) and q(J2) in G, respectively. Let V(q) be any parametric disk about q satisfying J1 n aV(q) # 0 and J2 n aV(q) # 0. Let J' and J' denote, respectively, the components of J1 n V(q) and J2 n V(q) which are not relatively compact on G. We say that q(J1) and q(J2) are identical when two points q1 E J' n aV(q) and q2 E= J5 n aV(q) can be joined by a Jordan arc J* c G n V(q). If not, then it is said that q(J1) and q(J2) are distinct. In this sense, let each point of aG be distinguished, and let h0(t) (0 < t < 1) denote a parametric representation of Cn. Let h be any bounded continuous real-valued function on aG U {p}. Since h h(p) is resolutive (cf. [1, Theorem 3.2]), h is resolutive (cf. [1, Theorem 8.1]). Therefore G* = G U aG u { p} is a resolutive compactification of G with respect to the relative topology of G* for R* (cf. [1, p. 87]). Henceforth we assume that p is regular with respect to G* in the sense of the Dirichlet problem, and that { p} is of harmonic measure 0 with respect to G*. In this paper we shall show the following Theorem and its applications. THEOREM. Let f be a bounded holomorphic function on G which is continuous Received by the editors April 6, 1976. AMS (MOS) subject classifications (1970). Primary 30A72; Secondary 30A50.
Starting Page	320
Ending Page	322
Page Count	3
File Format	PDF HTM / HTML
DOI	10.1090/S0002-9939-1976-0447561-0
Volume Number	61
Alternate Webpage(s)	http://www.ams.org/journals/proc/1976-061-02/S0002-9939-1976-0447561-0/S0002-9939-1976-0447561-0.pdf
Alternate Webpage(s)	https://doi.org/10.1090/S0002-9939-1976-0447561-0
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article