View source for Lindelöf theorem

''on asymptotic values''

1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590001.png" /> be a bounded regular analytic function in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590002.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590003.png" /> be the [[Asymptotic value|asymptotic value]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590004.png" /> along a Jordan arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590005.png" /> situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590006.png" /> and ending at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590007.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590008.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l0590009.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900010.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900011.png" /> is the [[Angular boundary value|angular boundary value]] (non-tangential boundary value) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900013.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900014.png" /> tends uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900015.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900016.png" /> inside an angle with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900017.png" /> formed by two chords of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900018.png" />.

The Lindelöf theorem is also true in domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900019.png" /> of other types, and the conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900020.png" /> have been significantly weakened. For example, it is sufficient to require that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900021.png" /> is a meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900022.png" /> that does not assume three different values. Lindelöf's theorem can also be generalized to functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900023.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900024.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900025.png" /> is a bounded holomorphic function in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900026.png" /> that has asymptotic value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900027.png" /> along a non-tangential path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900028.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900030.png" /> is the non-tangential boundary value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900031.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900032.png" /> (see [[#References|[4]]]).

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900033.png" /> be a bounded regular analytic function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900034.png" /> that has asymptotic values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900036.png" /> along two distinct paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900038.png" /> that end at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900039.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900041.png" /> uniformly inside the angle between the paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900043.png" />. This theorem is also true for domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900044.png" /> of other types. For unbounded functions it is false, generally speaking.

These theorems were discovered by E. Lindelöf [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.F. Collingwood,   A.J. Lohwater,   "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
For the generalization of Lindelöf's theorem to functions of several variables, the condition that the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900045.png" /> is non-tangential may be weakened, see [[#References|[a1]]], Chapt. 8.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059000/l05900046.png" />" , Springer  (1980)</TD></TR></table>