Cluster Sets by Kiyoshi Noshiro (auth.)

By Kiyoshi Noshiro (auth.)

For the 1st systematic investigations of the idea of cluster units of analytic services, we're indebted to IVERSEN [1-3J and GROSS [1-3J approximately 40 years in the past. next very important contributions sooner 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 eastern mathematicians; starting approximately 1940 a few contributions have been made to the idea via 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 way of BAGEMIHL, SEIDEL, COLLINGWOOD, CARTWRIGHT, HERVE, LEHTO, LOHWATER, MEIER, OHTSUKA and plenty of different mathematicians. the most goal of this small publication is to provide a scientific account at the conception of cluster units. bankruptcy I is dedicated to a couple definitions and initial discussions. In bankruptcy II, we deal with extensions of classical effects on cluster units to the case of single-valued analytic features in a common aircraft area whose boundary incorporates a compact set of crucial singularities of capability 0; it really is recognized that HALLSTROM [2J and TsuJI [7J prolonged independently Nevanlinna's concept of meromorphic features to the case of a compact set of crucial singUlarities of logarithmic potential 0. the following, Ahlfors' concept of overlaying surfaces performs a funda­ psychological function. bankruptcy III "is serious about features meromorphic within the unit circle.

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Their methods are based on the integral representation for a function J(z) of class (U). Suppose, for simplicity, that J(O) =l= 0 and denote its zero-points (if they exist) by av a2 , ••• , ak , • • • • Obviously I; (1-[a k ll < 00. We form the k Blaschke produkt B(z) = II k Uk iaki ak- z 1-ak z which is a function of class (U) 2. If we define g (z) by the relation J(z) = B (z) g (z), then g (z) is also a function of class (U) and does not vanish in [z[ < 1. Consider now the function h(z) = -logg(z), selecting a definite branch of the logarithm.

We define a f(z) - 1 Remark that 1 - 2 No SHIRO [11]. Cf. 3 HossJER OHTSUKA [2], [1]. r:J.. r:J.. f(z) is a function of class (U). also LOHWATER KAWAKAMI TSUJI [17]. [1]. [2]. KAMETANI-UGAERI [1]. LOHWATER-SEIDEL [1]. § 1. Functions of class (U) in Seidel's sense 35 boundary function V(8) in the following way: If an < 8 < bn, V(8) =cn 1fJn(8)+dn, where 1fJn(8) = [(b n -8) (8-an)J-t, and if eioEE, V(8) = +00, where two sequences of positive numbers {c n }, {dn } are 00 bn 00 chosen so that}; Cn J 1fJn(8) d8 < + 00, }; dn(bn- an) < +00 and dn n=i n=i an increases monotonically to infinity.

Theorem 12. (J. Furthermore, let the set E oj eiB E ~ E with ex E CAB (j, eiO ) be oj linear measure zero. Then, iJ ex is an exceptional value oj J(z) in a neighborhood oj zo, ex is an asymptotic value oj J(z) arbitrarily near ZO (NOSHIRO [I1J). Proof. It is obvious that Cf,-E (j, zo) C H. Since CD (j, zo) - C;; -E (j, zo) is an open set, Q 1 C CD (j, zo). We select a sufficiently small positive number r such that J(z) =1= ex in the intersection of D and (K): Iz - zo[ < r. , Z2= eiO , ((}1 < (}o < (}2) belonging to (~E - E) (\ (K).

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