# Lindelöf theorem

on asymptotic values

1) Let $w = f ( z)$ be a bounded regular analytic function in the unit disc $D = \{ {z } : {| z | < 1 } \}$ and let $\alpha$ be the asymptotic value of $f ( z)$ along a Jordan arc $L$ situated in $D$ and ending at a point $e ^ {i \theta _ {0} }$, that is, $f ( z) \rightarrow \alpha$ as $z \rightarrow e ^ {i \theta _ {0} }$ along $L$. Then $\alpha$ is the angular boundary value (non-tangential boundary value) of $f ( z)$ at $e ^ {i \theta _ {0} }$, that is, $f ( z)$ tends uniformly to $\alpha$ as $z \rightarrow e ^ {i \theta _ {0} }$ inside an angle with vertex $e ^ {i \theta _ {0} }$ formed by two chords of the disc $D$.

The Lindelöf theorem is also true in domains $D$ of other types, and the conditions on $f ( z)$ have been significantly weakened. For example, it is sufficient to require that $f ( z)$ is a meromorphic function in $D$ that does not assume three different values. Lindelöf's theorem can also be generalized to functions $f ( z)$ of several complex variables $z = ( z _ {1} \dots z _ {n} )$. For example, if $f ( z)$ is a bounded holomorphic function in the ball $D = \{ {z } : {| z | < 1 } \}$ that has asymptotic value $\alpha$ along a non-tangential path $L$ at a point $\zeta \in \partial D$, then $\alpha$ is the non-tangential boundary value of $f ( z)$ at $\zeta$( see [4]).

2) Let $w = f ( z)$ be a bounded regular analytic function in the disc $D = \{ {z } : {| z | < 1 } \}$ that has asymptotic values $\alpha$ and $\beta$ along two distinct paths $L _ {1}$ and $L _ {2}$ that end at the point $e ^ {i \theta _ {0} }$. Then $\alpha = \beta$ and $f ( z) \rightarrow \alpha$ uniformly inside the angle between the paths $L _ {1}$ and $L _ {2}$. This theorem is also true for domains $D$ 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

For the generalization of Lindelöf's theorem to functions of several variables, the condition that the path $L$ is non-tangential may be weakened, see [a1], Chapt. 8.