Sequences, wedges and associated sets of complex numbers. (English) Zbl 0659.30003

Hershkowitz, Daniel; Schneider, Hans

Czech. Math. J. 38(113), No. 1, 138-145 (1988).

Let \(R=r_ 1<r_ 2<..\). be an increasing sequence of positive integers and W(\(\alpha)\) (W[\(\alpha\) ]) the open (closed) wedge of width \({\mathcal L}^{\alpha}\) symmetrically located around the nonnegative real axis. In what follows n denotes a positive integer or \(n=\infty\). Denote by S(R,\(\alpha\),n) (S[R,\(\alpha\),n]) the set of all nonzero complex numbers c for which \(c^{r_ k}\in W(\alpha)\) \((c^{r_ k}\in W[\alpha])\) \((k=1,2,...,n)\). In the paper some sufficient and some necessary conditions for \(W(\alpha /r_ n)=S(R,\alpha,n)\) are established. Further a characterization for \(S(R,\alpha,n)\subseteq W(\alpha /r_{n-1})\) is given. Let T be a set of complex numbers, \(0\leq \alpha \leq \pi\). The sequence R is said to be a (T,\(\alpha)\)-forcing ((T,\(\alpha)\)- semiforcing) sequence provided that \(T\cap S[R,\alpha,| R|]\subseteq R_+\) \((T\cap S(R,\alpha,| R|)\subseteq R_+)\), where \(R_+=(0,+\infty)\) and \(| R|\) denotes the cardinality of the sequence R. Several sufficient conditions for R be a (T,\(\alpha)\)-(semi) forcing sequence are established in the paper.

Reviewer: T.Šalát

MSC:

30A99

General properties of functions of one complex variable

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