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 " , Springer (1981)
[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)