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 " , Springer (1980)