Kiyoshi Noshiro (auth.)'s Cluster Sets PDF

By Kiyoshi Noshiro (auth.)

ISBN-10: 3540025162

ISBN-13: 9783540025160

ISBN-10: 3642859283

ISBN-13: 9783642859281

For the 1st systematic investigations of the speculation of cluster units of analytic features, 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 through SEIDEL [1-2J, DOOE [1-4J, CARTWRIGHT [1-3J and BEURLING [1]. The investigations of SEIDEL and BEURLING gave nice impetus and curiosity to jap mathematicians; starting approximately 1940 a few contributions have been made to the speculation by way of 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 lots of different mathematicians. the most function of this small e-book is to offer a scientific account at the thought 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 capabilities in a basic aircraft area whose boundary encompasses a compact set of crucial singularities of ability 0; it really is famous that HALLSTROM [2J and TsuJI [7J prolonged independently Nevanlinna's thought of meromorphic services to the case of a compact set of crucial singUlarities of logarithmic means 0. right here, Ahlfors' concept of masking surfaces performs a funda­ psychological position. bankruptcy III "is serious about features meromorphic within the unit circle.

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156) raised a question whether this theorem holds in the case of a meromorphic function of unbounded type. We now discuss on this problem, although it is difficult to give a complete answer. With a slight modification of the proof of Theorem 12, we can prove Theorem 16. Let J(z) be meromorphic in Izl < 1. Suppose that CAO(j, eiO ) Iwl = 1 for almost every eiO belonging to an open arc A of lies on containing zo= eiOo , where Ao is an arbitrary curve in Izi < 1 ending at eiO . Let J(z) have a singularity at ZOo IJ J(z) has an exceptional value a (Ial = 1) in a neighborhood of ZO' then a is an asymptotic value along a path terminat- rw: r ing at some point arbitrarily near zO' under an additional condition that (i) tff = {e iO ; a~ CAO(j, eiO )} is everywhere dense on A.

156) raised a question whether this theorem holds in the case of a meromorphic function of unbounded type. We now discuss on this problem, although it is difficult to give a complete answer. With a slight modification of the proof of Theorem 12, we can prove Theorem 16. Let J(z) be meromorphic in Izl < 1. Suppose that CAO(j, eiO ) Iwl = 1 for almost every eiO belonging to an open arc A of lies on containing zo= eiOo , where Ao is an arbitrary curve in Izi < 1 ending at eiO . Let J(z) have a singularity at ZOo IJ J(z) has an exceptional value a (Ial = 1) in a neighborhood of ZO' then a is an asymptotic value along a path terminat- rw: r ing at some point arbitrarily near zO' under an additional condition that (i) tff = {e iO ; a~ CAO(j, eiO )} is everywhere dense on A.

Here 'Y) (0) and y (0) are non-decreasing; 'Y) (0) is continuous and 'Y)' (0) = 0 almost everywhere in 0 ~ 0 ~ 2n and y (0) a step-function. If 'Y) (0) is not constant, the symmetric derivative of 'Y) (0) is + 00 at a non-enumerable set of points!. Using these facts, we prove that there exists a point eirp for which the radial limit of g (z) is equal to zero. If 10 - tpl ~ 1- r = 0, then, by an elementary calculation, ~ ele + z ele-z - 1 - r2 I-2rcos(O-rp)+r2 Accordingly, f > 1 I-r 1 (z = rei rp) --;r' .

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Cluster Sets by Kiyoshi Noshiro (auth.)


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