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In [[complex analysis]], a branch of [[mathematics]], the '''Koebe 1/4 theorem''' states the following:

<blockquote>'''Koebe Quarter Theorem.''' the image of an injective analytic function ''f'' : '''D''' → '''C''' from the [[unit disk]] '''D''' onto a [[subset]] of the [[complex plane]] contains the disk whose center is ''f''(0) and whose radius is |''f′''(0)|/4.</blockquote>

The theorem is named after [[Paul Koebe]], who conjectured the result in 1907. The theorem was proven by [[Ludwig Bieberbach]] in 1916. The example of the Koebe function shows that the constant 1/4 in the theorem cannot be improved.

A related result is the [[Schwarz lemma]], and a notion related to both is [[conformal radius]].

==Gronwall's area theorem==
Suppose that

:<math>g(z) = z +b_1z^{-1} + b_2 z^{-2} + \cdots</math>

is univalent in |''z''| > 1. Then

:<math>\sum_{n\ge 1} n|b_n|^2 \le 1.</math>

In fact, if ''r'' > 1, the complement of the image of the disk ''|z|'' > ''r'' is a bounded domain ''X''(''r''). Its area is given by

:<math> \int_{X(r)} dxdy = {1\over 2i} \int_{\partial X(r)}\overline{z}\,dz = -{1\over 2i}\int_{|z|=r}\overline{g}\,dg={1\over 2\pi r^2} -{1\over 2\pi}\sum n|b_n|^2 r^{2n}.</math>

Since the area is positive, the result follows by letting ''r'' decrease to 1. The above proof shows equality holds if and only if the complement of the image of ''g'' has zero area, i.e. [[Lebesgue measure]] zero.

This result was proved in 1914 by the Swedish mathematician [[Thomas Hakon Gronwall]].

==Koebe function==
The '''Koebe function''' is defined by

:<math>f(z)=\frac{z}{(1 - z)^2}=\sum_{n=1}^\infty n z^n</math>

Application of the theorem to this function shows that the constant 1/4 in the theorem cannot be improved, as the image domain ''f''('''D''') does not contain the point ''z'' = −1/4 and so cannot contain any disk centred at 0 with radius larger than 1/4.

The '''rotated Koebe function''' is

:<math>f_\alpha(z)=\frac{z}{(1-\alpha z)^2}=\sum_{n=1}^\infty n\alpha^{n-1} z^n</math>

with α a complex number of [[absolute value]] 1. The Koebe function and its rotations are ''[[Schlicht function|schlicht]]'': that is, [[Univalent function|univalent]] (analytic and [[Injective function|one-to-one]]) and satisfying ''f''(0) = 0 and ''f′''(0) = 1.

==Bieberbach's coefficient inequality for univalent functions==
Let

:<math> g(z) = z + a_2z^2 + a_3 z^3 + \cdots </math>

be univalent in |''z''| < 1. Then

:<math>|a_2|\le 2.</math>

This follows by applying Gronwall's area theorem to the odd univalent function

:<math> g(z^2)^{-1/2}= z^{-1} -{1\over 2} a_2 z + \cdots. </math>

Equality holds if and only if ''g'' is a rotated Koebe function.

This result was proved by [[Ludwig Bieberbach]] in 1916 and provided the basis for his [[Bieberbach conjecture|celebrated conjecture]] that |''a''<sub>''n''</sub>| ≤ ''n'', proved in 1985 by [[Louis de Branges]].

==Proof of quarter theorem==
Applying an affine map, it can be assumed that

:<math>f(0)=0,\,\,\, f^\prime(0)=1,</math>

so that

:<math> f(z) = z + a_2 z^2 + \cdots .</math>

If ''w'' is not in ''f''('''D'''), then

:<math>h(z)={wf(z)\over w-f(z)} = z +(a_2+w^{-1}) z^2 + \cdots </math>

is univalent in |''z''| < 1.

Applying the coefficient inequality to ''f'' and ''h'' gives

:<math> |w|^{-1} \le |a_2| + |a_2 + w^{-1}|\le 4, </math>

so that

:<math> |w|\ge {1\over 4}.</math>

==Koebe distortion theorem==
The '''Koebe distortion theorem''' gives a series of bounds for a univalent function and its derivative. It is a direct consequence of Bieberbach's inequality for the second coefficient and the Koebe quarter theorem.<ref>{{harvnb|Pommerenke|1975|pp=21–22}}</ref>

Let ''f''(''z'') be a univalent function on |''z''| < 1 normalized so that ''f''(0) = 0 and ''f'''(0) = 1 and let ''r'' = |''z''|. Then

:<math>{r \over (1+r)^2}\le |f(z)|\le {r\over (1-r)^2}</math>

:<math>{1-r\over (1+r)^3} \le |f^\prime(z)| \le {1+r\over (1-r)^3}</math>

:<math>{1-r\over 1+r} \le \left|z{f^\prime(z)\over f(z)}\right| \le {1+r\over 1-r}</math>

with equality if and only if ''f'' is a Koebe function

:<math> f(z) ={z\over(1-e^{i\theta}z)^2}. </math>

==Notes==
{{reflist|2}}

==References==
*{{citation|first=Ludwig|last=Bieberbach|title=Über die Koeffizienten derjenigen Polenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln|journal= S.-B. Preuss. Akad. Wiss.|year= 1916|pages= 940–955}}
*{{citation|last=Carleson|first=L.|last2= Gamelin|first2= T. D. W.|title=Complex dynamics|series=Universitext: Tracts in Mathematics|publisher= Springer-Verlag|year=1993|isbn=0-387-97942-5|pages=1–2}}
* {{Citation | last1=Conway | first1=John B. | author1-link=John B. Conway | title=Functions of One Complex Variable II | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94460-9 | year=1995}}
*{{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}}
*{{citation|first=T.H.|last=Gronwall|title=Some remarks on conformal representation|journal= Ann. of Math.|volume= 16 |year=1914|pages= 72–76}}
*{{citation|first=Zeev|last=Nehari|authorlink=Zeev Nehari|title=Conformal mapping|publisher=Dover|year=1952|isbn=0-486-61137-X|pages=248–249}}
*{{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}}
*{{cite book| last=Rudin | first=Walter | authorlink=Walter Rudin | year=1987 | title=Real and Complex Analysis | series=Series in Higher Mathematics | publisher=McGraw-Hill | edition=3 | isbn=0-07-054234-1 | mr=924157}}

==External links==
* Koebe 1/4 theorem at [http://planetmath.org/encyclopedia/KobeOneFourthTheorem.html PlanetMath]

[[Category:Theorems in complex analysis]]

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