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In [[mathematics]], '''Grunsky's theorem''', due to the German mathematician [[Helmut Grunsky]], is a result in [[complex analysis]] concerning [[holomorphic]] [[univalent function]]s defined on the [[unit disk]] in the [[complex numbers]]. The theorem  states that a univalent function defined on the unit disc, fixing the point 0, maps every disk ''|z|'' < ''r'' onto a [[star domain|starlike domain]] for ''r'' ≤ tanh π/4. The largest ''r'' for which this is true is called the '''radius of starlikeness''' of the function.


==Statement of theorem==
Let ''f'' be a univalent holomorphic function on the unit disc ''D'' such that ''f''(0) = 0. Then for all ''r'' ≤ tanh&nbsp;π/4, the image of the disc ''|z|'' < ''r'' is [[star domain|starlike]] with respect to 0, , i.e. it is invariant under multiplication by real numbers in (0,1).


==An inequality of Grunsky==
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If ''f''(z) is univalent on ''D'' with ''f''(0) = 0, then
 
:<math>\left|\log {zf^\prime(z)\over f(z)}\right|\le \log {1+|z|\over 1-|z|}.</math>
 
Taking the real and imaginary parts of the logarithm, this implies the two inequalities
 
:<math>\left|{zf^\prime(z)\over f(z)}\right|\le {1+|z|\over 1-|z|}</math>
 
and
 
:<math>\left|\arg {zf^\prime(z)\over f(z)}\right| \le \log {1+|z|\over 1-|z|}.</math>
 
For fixed ''z'', both these equalities are attained by suitable [[Koebe function]]s
 
:<math> g_w(\zeta)={\zeta\over (1-\overline{w}\zeta)^2},</math>
 
where ''|w|'' = 1.
 
===Proof of inequality===
{{harvtxt|Grunsky|1932}} originally proved these inequalities based on extremal techniques of [[Ludwig Bieberbach]]. Subsequent proofs, outlined in {{harvtxt|Goluzin|1939}}, relied on the [[Loewner equation]]. More elementary proofs were subsequently given based on [[Goluzin's inequalities]], an equivalent form of [[Grunsky's inequalities]] (1939) for the [[Grunsky matrix]].
 
For a univalent function ''g''  in ''z'' > 1 with an expansion
 
:<math> g(z) = z + b_1 z^{-1} + b_2 z^{-2} + \cdots.</math>
 
Goluzin's inequalities state that
 
:<math> \left|\sum_{i=1}^n\sum_{j=1}^n\lambda_i\lambda_j \log {g(z_i)-g(z_j)\over z_i-z_j}\right| \le \sum_{i=1}^n\sum_{j=1}^n \lambda_i\overline{\lambda_j}\log {z_i\overline{z_j}\over z_i\overline{z_j}-1},</math>
 
where the ''z''<sub>''i''</sub> are distinct points with |''z''<sub>''i''</sub>| > 1  and λ<sub>''i''</sub> are arbitrary complex numbers.
 
Taking ''n'' = 2. with λ<sub>''1''</sub> = – λ<sub>''2''</sub> = λ, the inequality implies
 
:<math> \left| \log {g^\prime(\zeta)g^\prime(\eta) (\zeta-\eta)^2\over (g(\zeta)-g(\eta))^2}\right|\le \log {|1-\zeta\overline{\eta}|^2\over (|\zeta|^2 -1 )(|\eta|^2 -1)}.</math>
 
If ''g'' is an odd function and η =  – ζ, this yields
 
:<math> \left| \log {\zeta g^\prime(\zeta) \over g(\zeta)}\right| \le {|\zeta|^2 + 1\over |\zeta|^2 -1}.</math>
 
Finally if ''f'' is any normalized univalent function in ''D'', the required inequality for ''f'' follows by taking
 
:<math> g(\zeta)=f(\zeta^{-2})^{-{1\over 2}}</math>
 
with <math>z=\zeta^{-2}.</math>
 
==Proof of theorem==
Let ''f'' be a univalent function on ''D'' with ''f''(0) = 0. By [[Nevanlinna's criterion]],  ''f'' is starlike on ''|z|'' < ''r'' if and only if
 
:<math> \Re {zf^\prime(z)\over f(z)} \ge 0</math>
 
for ''|z|'' < ''r''. Equivalently
 
:<math>\left|\arg {zf^\prime(z)\over f(z)}\right| \le {\pi\over 2}.</math>
 
On the other hand by the inequality of Grunsky above,
 
:<math> \left|\arg {zf^\prime(z)\over f(z)}\right|\le \log {1+|z|\over 1-|z|}.</math>
 
Thus if
 
:<math> \log {1+|z|\over 1-|z|} \le  {\pi\over 2},</math>
 
the inequality holds at ''z''. This condition is equivalent to
 
:<math>|z|\le \tanh {\pi\over 4} </math>
 
and hence ''f'' is starlike on any disk ''|z|'' < ''r'' with ''r'' ≤ tanh π/4.
 
==References==
*{{citation|last=Duren|first=P. L.|title=
Univalent functions|series=Grundlehren der Mathematischen Wissenschaften|volume= 259|publisher= Springer-Verlag|year= 1983|isbn= 0-387-90795-5|pages=95–98}}
*{{citation|last=Goluzin|first=G.M.|journal=Uspekhi Mat. Nauk|year= 1939|volume= 6|pages=26–89|title=Interior problems of the theory of univalent functions|
url= http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=8936&option_lang=eng}} (in Russian)
*{{citation|last=Goluzin|first= G. M.|title=Geometric theory of functions of a complex variable|series=Translations of Mathematical Monographs|volume=26| publisher=American Mathematical Society|year= 1969}}
*{{citation|first=A.W.|last=Goodman|title=Univalent functions|publisher=Mariner Publishing Co.|year= 1983|volume=I|isbn=0-936166-10-X}}
*{{citation|first=A.W.|last=Goodman|title=Univalent functions|publisher=Mariner Publishing Co.|year= 1983|volume=II|isbn=0-936166-11-8}}
*{{citation|first=H.|last=Grunsky|title=Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche (inaugural dissertation)|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN322068231|year=1932|volume=1|journal=Schr. Math. Inst. u. Inst. Angew. Math. Univ. Berlin|pages=95–140}} (in German)
*{{citation|first=H.|last=Grunsky|title=Zwei Bemerkungen zur konformen Abbildung|journal=Jber. Deutsch. Math.-Verein.|volume= 43 |year=1934|pages=140–143|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002130416}} (in German)
*{{citation|last=Hayman|first= W. K.|title=Multivalent functions|edition=2nd|series=Cambridge Tracts in Mathematics|volume=110|publisher= Cambridge University Press|year= 1994|isbn= 0-521-46026-3}}
*{{citation|last=Nevanlinna|first= R.|title=Über die konforme Abbildung von Sterngebieten|journal=Ofvers. Finska Vet. Soc. Forh. |volume=53 |year=1921|pages=1–21}}
*{{citation|last=Pommerenke|first= C.|authorlink=Christian Pommerenke|title=Univalent functions, with a chapter on quadratic differentials by Gerd Jensen|series= Studia Mathematica/Mathematische Lehrbücher|volume=15|publisher= Vandenhoeck & Ruprecht|year= 1975}}
 
[[Category:Theorems in complex analysis]]

Latest revision as of 17:48, 3 July 2014


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