Lindelöf theorem
on asymptotic values
1) Let be a bounded regular analytic function in the unit disc
and let
be the asymptotic value of
along a Jordan arc
situated in
and ending at a point
, that is,
as
along
. Then
is the angular boundary value (non-tangential boundary value) of
at
, that is,
tends uniformly to
as
inside an angle with vertex
formed by two chords of the disc
.
The Lindelöf theorem is also true in domains of other types, and the conditions on
have been significantly weakened. For example, it is sufficient to require that
is a meromorphic function in
that does not assume three different values. Lindelöf's theorem can also be generalized to functions
of several complex variables
. For example, if
is a bounded holomorphic function in the ball
that has asymptotic value
along a non-tangential path
at a point
, then
is the non-tangential boundary value of
at
(see [4]).
2) Let be a bounded regular analytic function in the disc
that has asymptotic values
and
along two distinct paths
and
that end at the point
. Then
and
uniformly inside the angle between the paths
and
. This theorem is also true for domains
of other types. For unbounded functions it is false, generally speaking.
These theorems were discovered by E. Lindelöf [1].
References
[1] | E. Lindelöf, "Sur un principe générale de l'analyse et ses applications à la théorie de la représentation conforme" Acta Soc. Sci. Fennica , 46 : 4 (1915) pp. 1–35 |
[2] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
[3] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6 |
[4] | E.M. [E.M. Chirka] Čirka, G.M. [G.M. Khenkin] Henkin, "Boundary properties of holomorphic functions of several complex variables" J. Soviet Math. , 5 (1976) pp. 612–687 Itogi Nauk. i Tekhn. Sovrem. Probl. , 4 (1975) pp. 13–142 |
Comments
For the generalization of Lindelöf's theorem to functions of several variables, the condition that the path is non-tangential may be weakened, see [a1], Chapt. 8.
References
[a1] | W. Rudin, "Function theory in the unit ball in ![]() |
Lindelöf theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_theorem&oldid=22759