Meier theorem

Let $f(z)$ be a meromorphic function in the unit disc $D = \{ z \in \mathbf{C} : |z| < 1 \}$; then all points of the circle $\Gamma = \{ z \in \mathbf{C} : |z| = 1 \}$ except, possibly, for a set of the first category on $\Gamma$, are either Plessner points or Meier points. By definition, a point $e^{i\theta}$ on $\Gamma$ is a Plessner point for $f$ if the angular cluster set $C_A(e^{i\theta},f)$ is total (i.e., coincides with the whole extended complex plane $\bar{\mathbf{C}}$) for every angle $\delta$ between pairs of chords through $e^{i\theta}$. The point $e^{i\theta}$ is said to be a Meier point (or to have the Meier property) if: 1) the complete cluster set $C(e^{i\theta},f)$ of $f$ at $e^{i\theta}$ is subtotal, i.e. does not coincide with the whole extended complex plane; and 2) the set of all limit values along arbitrary chords of the disc $D$ drawn at the point $e^{i\theta}$ is identical to $C(e^{i\theta},f)$. The theorem was proved by K. Meier [1].

Meier's theorem is the analogue, in terms of the category of a set, of the Plessner theorem, which is formulated in terms of measure theory. A sharpening of Meier's theorem is given in [3].

References

[1]	K. Meier, "Ueber die Randwerte der meromorphen Funktionen" Math. Ann , 142 (1961) pp. 328–344
[2]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6
[3]	V.I. Gavrilov, A.N. Kanatnikov, "Characterization of the set $M(f)$ for meromorphic functions" Soviet Math. Dokl. , 18 : 2 (1977) pp. 15–17 Dokl. Akad. Nauk SSSR , 233 : 1 (1977)