Meier theorem
Let be a meromorphic function in the unit disc
; then all points of the circle
except, possibly, for a set of the first category on
, are either Plessner points or Meier points. By definition, a point
on
is a Plessner point for
if the angular cluster set
is total (i.e., coincides with the whole extended complex plane
) for every angle
between pairs of chords through
. The point
is said to be a Meier point (or to have the Meier property) if: 1) the complete cluster set
of
at
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
drawn at the point
is identical to
. 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 ![]() |
Meier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meier_theorem&oldid=41112