View source for Julia-Wolff-Carathéodory theorem

''Julia–Carathéodory theorem, Julia–Wolff theorem''

A classical statement which combines the celebrated [[Julia theorem|Julia theorem]] from 1920 [[#References|[a18]]], Carathéodory's contribution from 1929 [[#References|[a7]]] (see also [[#References|[a8]]]), and Wolff's boundary version of the [[Schwarz lemma|Schwarz lemma]] from 1926 [[#References|[a30]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300701.png" /> be the open unit disc in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300702.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300703.png" /> be the set of all holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300704.png" /> with values in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300705.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300706.png" /> (cf. also [[Analytic function|Analytic function]]). For the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300707.png" />, of holomorphic self-mappings on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300708.png" />, one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j1300709.png" />; it is a [[Semi-group of holomorphic mappings|semi-group of holomorphic mappings]] with respect to composition.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007010.png" /> on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007011.png" />, the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007013.png" />, a non-tangential approach region at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007014.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

The term  "non-tangential"  refers to the fact that at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007017.png" /> lies in the sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007018.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007019.png" />) that is the region between two straight lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007020.png" /> meeting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007021.png" /> and symmetric about the radius to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007022.png" />, the boundary curves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007023.png" /> having a corner at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007024.png" />, with angle less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007025.png" />.

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007026.png" /> is said to have a non-tangential limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007027.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007029.png" /> exists in each non-tangential region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007030.png" />. In this case one also writes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007031.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007033.png" />, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007035.png" />, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007037.png" /> is a closed disc internally tangent to the circle at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007038.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007039.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007040.png" />. Such a disc is called a horodisc (cf. also [[Horosphere|Horosphere]]).

In 1920, G. Julia [[#References|[a18]]] identified hypotheses showing how to get the existence of the non-tangential limit at a given boundary point.

==Julia's lemma.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007041.png" /> be not constant. Suppose that there are points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007043.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007044.png" />, such that for a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007045.png" /> converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007046.png" /> the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007047.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007048.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

Then

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007050.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007051.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007052.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007053.png" />;

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007054.png" /> exists and is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007055.png" />. Moreover, if the equality in ii) holds for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007057.png" /> is an automorphism of the disc.

==Julia–Carathéodory theorem.==
In 1929, C. Carathéodory [[#References|[a7]]] proved that under Julia's hypotheses the derivative also admits a non-tangential limit at the same boundary point.

Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007058.png" />. Then the following statements are equivalent:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007059.png" />, where the limit is taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007060.png" /> approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007061.png" /> unrestrictedly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007062.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007063.png" /> exists for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007064.png" />;

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007065.png" /> exists, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007066.png" />. Moreover,

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007067.png" /> in i);

b) the boundary points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007068.png" /> in ii) and iii) are the same;

c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007069.png" />.

After appropriate preliminary rotations, one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007070.png" />. Thus, these results show that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007071.png" /> has an angular derivative at some boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007072.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007073.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007074.png" /> cannot have an interior fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007075.png" />.

Now assume only that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007076.png" /> has no interior fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007077.png" />. The question then is: Does the angular derivative at a certain point on the boundary exist?

The affirmative answer was given by J. Wolff [[#References|[a30]]] in 1926.

==Wolff's theorem.==
Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007078.png" /> has no fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007079.png" />. Then there is a unique unimodular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007080.png" /> such that

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007081.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007082.png" />;

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007083.png" /> exists and is less than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007084.png" />.

The latter assertion can be interpreted as a direct analogue of the Schwarz–Pick lemma (cf. [[Schwarz lemma|Schwarz lemma]]), where the role of the fixed point is taken over by a point on the unit circle. Moreover, this result is the key to all the deeper facts about sequences of iterates.

==Generalizations.==
There are various versions and proofs of the Julia–Carathéodory theorem (sometimes also called the Julia–Wolff–Carathéodory theorem or Julia–Wolff theorem). For the one-dimensional case, see, for example, [[#References|[a19]]], [[#References|[a29]]], [[#References|[a27]]], [[#References|[a12]]], [[#References|[a26]]], [[#References|[a6]]], [[#References|[a21]]] or [[#References|[a8]]], [[#References|[a22]]], [[#References|[a28]]], [[#References|[a9]]].

Note that D. Sarason [[#References|[a26]]] gave an interesting proof of the Julia–Carathéodory theorem by using Hilbert space constructions for angular derivatives.

A strengthened version of Julia's lemma was established by P.R. Mercer [[#References|[a21]]], employing techniques for the hyperbolic Poincaré metric (cf. [[Poincaré model|Poincaré model]]).

Different generalizations of the Julia–Wolff–Carathéodory theorem for bounded domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007085.png" /> are known: for the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007086.png" /> ([[#References|[a16]]], [[#References|[a25]]]), for the poly-disc ([[#References|[a17]]], [[#References|[a4]]]), for strongly convex and strongly pseudo-convex domains ([[#References|[a2]]], [[#References|[a3]]]). Also, M. Abate and R. Tauraso [[#References|[a1]]] have described a general framework allowing one to generalize the Julia–Wolff–Carathéodory theorem in terms of the Kobayashi metric (cf. also [[Hyperbolic metric|Hyperbolic metric]]; [[Kobayashi hyperbolicity|Kobayashi hyperbolicity]]) on a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007087.png" />.

For generalizations of Wolff's theorem in the unit ball of a complex Hilbert space, see [[#References|[a13]]] and [[#References|[a14]]].

Earlier, V.P. Potapov [[#References|[a23]]] extended Julia's lemma to matrix-valued holomorphic mappings of a complex variable. His results, as well as the Julia–Wolff–Carathéodory theorem, were generalized by K. Fan and T. Ando ([[#References|[a10]]], [[#References|[a11]]] and [[#References|[a5]]]) to operator-valued holomorphic mappings. Also, in these works they extended the Julia–Wolff–Carathéodory theorem to holomorphic mappings of proper contractions on the unit Hilbert ball acting in the sense of functional calculus.

K. Wlodarczyk [[#References|[a31]]] and P. Mellon [[#References|[a20]]] have presented some more general results in this direction for the holomorphic mappings on the open unit ball of so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130070/j13007089.png" />-algebras, using techniques developed by L.A. Harris [[#References|[a15]]].

For a survey of work in higher dimensions, see [[#References|[a25]]], [[#References|[a14]]], [[#References|[a9]]], [[#References|[a24]]], [[#References|[a20]]], [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Abate,   R. Tauraso,   "The Julia–Wolff–Caratheodory theorem(s)"  ''Contemp. Math.'' , '''222'''  (1999)  pp. 161–172</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abate,   "The Lindelöf principle and the angular derivative in strongly convex domains"  ''J. Anal. Math.'' , '''54'''  (1990)  pp. 189–228</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Abate,   "Angular derivatives in strongly pseudoconvex domains" , ''Proc. Symp. Pure Math.'' , '''52/2''' , Amer. Math. Soc.  (1991)  pp. 23–40</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Abate,   "The Julia–Wolff–Caratheodory theorem in polydisks"  ''J. Anal. Math.'' , '''74'''  (1998)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Ando,   K. Fan,   "Pick–Julia theorems for operators"  ''Math. Z.'' , '''168'''  (1979)  pp. 23–34</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.B. 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