Some notes on the cluster sets of meromorphic functions

Note on the cluster sets of meromorphic functions

“…We consider the open set of all points z in D Π (L) such that f(z) e (X, X') and we denote by Δ its component containing the path Λ. Denote by Δ r the common part of Δ and the exterior of y r , by A(r) the area of the Riemannian image of Δ r under the function w =f(z) and by L[r) the total length of the image of Δuγ r . By Matsumoto's argument [2], we can get easily the following lemma under the assumption of our Theorem. lim inf L(r)/A(r) = 0.…”

“…It is known that if E is of logarithmic capacity zero, then there exists an exhaustion of F satisfying (A), and that if there exists an exhaustion of F satisfying (A), then E belongs to the class N 0 B (cf. Matsumoto [2]). Matsumoto [2] gave an example of a Cantor set E which is of logarithmic capacity positive and whose complement has an exhaustion satisfying (A).…”

“…Matsumoto [2]). Matsumoto [2] gave an example of a Cantor set E which is of logarithmic capacity positive and whose complement has an exhaustion satisfying (A).…”

“…1. Herve [1] and Matsumoto [2] have proved the interesting theorems on cluster sets of single-valued meromorphic functions in general domains.…”