Non-Essential Cluster Values and Normal Functions

Abstract

We consider continuous functions f which map the open unit disk D into the Riemann sphere W. For a point ζ on the unit circle C, we say that χ is a chord at ζ if χ is a chord of C having one endpoint at ζ and that Δ is a Stolz angle at ζ if Δ is a Stolz angle with vertex ζ. Suppose S denotes either a chord at ζ, a Stolz angle at ζ, or D.

Cluster sets and essential cluster sets of meromorphic functions.

“…Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/31/15 7:18 AM From Theorem l, Theorem 3 and the following Lemma 4, we can easily obtain the following Theorem 4 which is the same s a result of Belna [3], Theorem 3.…”

“…Corollary 2 is equal to a result of Belna [3], p. 52, Corollary 1. Corollary 2 is equal to a result of Belna [3], p. 52, Corollary 1.…”

“…In the case that f(z) is a continuous function in t/, the following Corollary 4 is analogous to a result of Belna [3], p. 54, Theorem 2. In the case that f(z) is a continuous function in t/, the following Corollary 4 is analogous to a result of Belna [3], p. 54, Theorem 2.…”

“…For normal functions in i/, Theorem 4 is equal to a result of Belna [3], Theorem 3. In section 3, we give some results relative to a Stolz angle (or an angular domain) analogous to those in section 2.…”

“…We denote the spherical derivative On the other band, from [1], Theorem l, we see that for the elliptic modular function f(z) defined in U, the set of points ζ of C satisfying C A (f, ζ) Φ W is of measure 0 and of the first Baire category on C, but N(f) = C. Hence, Theorem 7 is considerably sharper than those results of Belna [3], p. 52, Corollary 2, and p. 54, Corollary 2. We denote the spherical derivative On the other band, from [1], Theorem l, we see that for the elliptic modular function f(z) defined in U, the set of points ζ of C satisfying C A (f, ζ) Φ W is of measure 0 and of the first Baire category on C, but N(f) = C. Hence, Theorem 7 is considerably sharper than those results of Belna [3], p. 52, Corollary 2, and p. 54, Corollary 2.…”