By Kiyoshi Noshiro (auth.)
For the 1st systematic investigations of the speculation of cluster units of analytic capabilities, we're indebted to IVERSEN [1-3J and GROSS [1-3J approximately 40 years in the past. next vital contributions earlier than 1940 have been made via SEIDEL [1-2J, DOOE [1-4J, CARTWRIGHT [1-3J and BEURLING . The investigations of SEIDEL and BEURLING gave nice impetus and curiosity to eastern mathematicians; starting approximately 1940 a few contributions have been made to the idea through KUNUGUI [1-3J, IRIE [IJ, TOKI [IJ, TUMURA [1-2J, KAMETANI [1-4J, TsuJI [4J and NOSHIRO [1-4J. lately, many noteworthy advances were made by means of BAGEMIHL, SEIDEL, COLLINGWOOD, CARTWRIGHT, HERVE, LEHTO, LOHWATER, MEIER, OHTSUKA and plenty of different mathematicians. the most objective of this small publication is to provide a scientific account at the conception of cluster units. bankruptcy I is dedicated to a few definitions and initial discussions. In bankruptcy II, we deal with extensions of classical effects on cluster units to the case of single-valued analytic services in a common airplane area whose boundary encompasses a compact set of crucial singularities of skill 0; it really is famous that HALLSTROM [2J and TsuJI [7J prolonged independently Nevanlinna's idea of meromorphic capabilities to the case of a compact set of crucial singUlarities of logarithmic means 0. the following, Ahlfors' conception of overlaying surfaces performs a funda psychological position. bankruptcy III "is occupied with features meromorphic within the unit circle.
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Then, by Theorem 1, § 3, atl and at2 are asymptotic values of f(z) along two paths Al and A2 in D terminating at ZOo By a well-known theorem of IVERSEN-LINDELOF 2 on asymptotic values, w = f(z) takes every value infinitely often except perhaps two values between Al and A 2 • Accordingly, RD(j, zo) coincides with the domain obtained by excluding two points atl and at2 from the w-plane; hence CD (j, zo) is the whole w-plane. As an alternative, if CD (j, zo) is not the whole w-plane, every value of Q belongs to RD (j, zo) with one possible exception.
Thus E (G l ) = E (G 2 ). As an immediate consequence of this fact, if Zo is not an accumulation point of El and if Cr(j, zo) - Cr- E(j, zo) =F B, then this set is the union of some components Q n of Q = CD (j, Zo) - Cr- E(j, zo) and hence an open set. For if wE Cdf, zo) - Cr_E(j, zo), then any domain G which contains w in its interior and whose closure is contained in Qn' must be contained in Cr(j, zo) - Cr-E(j, zo)' Theorem 1. Let D be a domain, r its boundary, E a countable compact set contained in rand Zo a point of E.
Using the same notations in Paragraph 3, we start with the following Theorem 13. Suppose that jar every ei 0 E rc E = A - E, CA0 (j, ei 0) lies on the circumjerence rw: Iwl = 1 and that CDU, zo) = C'f-E(j, zo). Then, w = j(z) is regular at z = ZOo Proof. Obviously, j(z) is regular and bounded in the intersection of Izl < 1 with a neighborhood of zo; there is an open arc Al of A, containing Zo in its interior, such that there exists the radial limit j(eiO ) for almost every eiO E A I - E. Since there are at most countably many ambiguous points ofj(z) on Izl = 1, we have Ij(eiO)1 = 1 for almost every point eiO of A I - E.