The boundary values of a class of meromorphic functions

Lohwater, A. J.

doi:10.1215/s0012-7094-52-01925-x

Cited by 14 publications

(7 citation statements)

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“…Let us now consider the less restrictive case where ƒ is only assumed to have an asymptotic value at each point of a set on {I z\ = 1} of measure 27r. Let E be a closed subset of W such that for almost all e i$ , ƒ has an asymptotic value at e i9 that is in E. By Bagemihl's ambiguous-point theorem [l], FQE (also F C QE), and we see that our results contain some theorems of Lehto [4] and Lohwater [5]. Our results also contain theorems of Noshiro [8], Stoïlow [lO], and McMillan [6].…”

Section: Define Sets R(u) and F E As Follows: Wçzr(u) If And Only If supporting

confidence: 63%

On the geometric theory of functions meromorphic in a disc

McMillan¹

1968

Bull. Amer. Math. Soc.

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Let w = ƒ(z) be a nonconstant meromorphic function defined in the open unit disc D, and let W denote the extended ze/-plane. Let N(w, ô) denote the set of all points of W at a chordal distance less than S from w (ô>0), and define a closed set BQW as follows: w&B if and only if wÇzW and for any S>0 there exist N(wo, 8 0 ) CN(w, 5) and a component U of the preimage f"x (N(wo t ô 0 )) such that f(U) is not dense in N(WQ, S 0 ). THEOREM Suppose that Vis a domain contained in the complement of B and that U is a component off~~x(V). Then one of the following two statements holds: (i) For any w(EV there exists d>0 such that Ur\f" 1 (N(w 1 ô)) is relatively compact (in D).( ii) For any wÇzV either there exists a continuous curve a: z(t), Og*1 and f(z(t))-*w as /->1, or there exists ô>0 such that infinitely many relatively compact components of f~l(N(w, $)) are contained in U.The proofs of the results stated in this note will appear in a forthcoming paper.A point w(EW is called an asymptotic value of ƒ provided there exists a continuous curve a: z(t), Og/1 and f(z(t))-*w as J-»l, and if in addition zty-te* 0 , then ƒ is said to have the asymptotic value w at e ie . We use the definition of linear measure that is given in terms of coverings by discs. The measurability of A (N(w, 5)) and V p is proved in (7).

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Section: Define Sets R(u) and F E As Follows: Wçzr(u) If And Only If supporting

confidence: 63%

On the geometric theory of functions meromorphic in a disc

McMillan¹

1968

Bull. Amer. Math. Soc.

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“…Then, (4) Z/ y (0) Π 0 for i < J and -3/2 < 0 < 0. Thus, from (3) and (4) we have that the arc of \z\ -1 cut off by L 2 (θ) is greater than the arc of \z\ -1 cut off by Vβ) for j > J and -δ/2 < θ < 0. Thus, for all j and -δ/2 < 0 < 0.…”

Section: K = Lmentioning

confidence: 93%

On the Distribution of Values of Functions in the Unit Disk

Choike

1973

Nagoya Mathematical Journal

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Let f(z) be a function analytic and bounded, |f(z)| < 1, in |z| < 1. Then, by Fatou’s theorem the radial limit f*(eiθ) = limr→1f(reiθ) exists almost everywhere on |z| = 1. Seidel [8, p. 208] and Calderón, González- Domínguez, and Zygmund [1] (see also [9, pp. 281-282]) proved the following: if f*(eiθ) is of modulus 1 almost everywhere on an arc a < θ < b of |z| = 1, then either f(z) is analytically continuable across this arc or the values f*(eiθ), a < d < b, cover the circumference |w| = 1 infinitely many times.

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“…See for example the work of Lohwater [4], [5], [6], Noshiro [7], [8], Tsuji [11] and the author [10]. We now prove a lemma concerning the structure of G(αr, p, k) for functions of class (£7*).…”

Section: Definitionmentioning

confidence: 97%