A Boundary Theorem for Tsuji Functions

Abstract

Let D denote the unit disc | z | <1, C the unit circle | z | = 1 and Cr the circle | z| = r. Corresponding to any function w(z) meromorphic in D we denote by w*(z) the spherical derivative

Strongly Annular Tsuji Functions

“…Theorem 2 contradicts the statement at the end of page 199 of Collingwood's paper [3]. Collingwood stated that even if a Tsuji function has no Fatou points, the set of its Julia points may still be of first category.…”

“…E. F. Collingwood [3] proved that if f is a meromorphic Tsuji function in D, then almost every point on the unit circle C is either a Fatou point or a Julia point for f. Because the Fatou points of an annular function constitute a set of measure 0, the set of Julia points of an annular Tsuji function must have measure In. That it is also a residual subset of C follows immediately from our second theorem.…”

“…The proposed proof is based on an example furnished by Piranian in an informal communication. Piranian regrets that his construction involves an error: the first clause in the final sentence (middle of page 200) in [3] is incorrect.…”