Plessner theorem
One of the basic results in the theory of boundary properties of analytic functions. Let be a meromorphic function in the unit disc
and let
be the open angle with vertex
on the circle
formed by two chords of
passing through
. The point
is called a Plessner point (or it is said that
has the Plessner property) if in every arbitrarily small angle
there exists a sequence
such that
![]() |
for every value in the extended complex plane
. The point
is called a Fatou point for
if there exists a single unique limit
![]() |
as tends to
within any angle
. Plessner's theorem [1]: Almost-all points on
with respect to the Lebesgue measure on
are either Fatou points or Plessner points.
It is also known that the set of all Plessner points has type
on
. Examples have been constructed of analytic functions in
for which
is dense on
and has arbitrary given Lebesgue measure
,
[3]. Plessner's theorem applies to any meromorphic function
in any simply-connected domain
with a rectifiable boundary
. In that case,
is a Fatou point if the following limit exists (cf. also Cluster set):
![]() |
as along any non-tangential path; the definition of a Plessner point
must be altered in such a way that one considers angles
with vertex
and sides forming angles less than
with the normal to
at
[2].
Meier's theorem is an analogue of Plessner's theorem in terms of categories of sets (cf. Meier theorem).
References
[1] | A.I. Plessner, "Über das Verhalten analytischer Funktionen auf dem Rande des Definitionsbereiches" J. Reine Angew. Math. , 158 (1928) pp. 219–227 |
[2] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |
[3] | A.J. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian) |
Comments
Angles of the form are called Stolz angles.
A good reference is [a3], to which [3] is a Russian sequel.
Plessner's theorem has a complete analogue for the unit ball in , cf. [a1]: Every holomorphic function on the unit ball decomposes the boundary into three measurable sets, as in the classical case.
References
[a1] | W. Rudin, "Function theory in the unit ball in ![]() |
[a2] | M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975) |
[a3] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9 |
[a4] | K. Noshiro, "Cluster sets" , Springer (1960) |
Plessner theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plessner_theorem&oldid=48190