Koebe quarter theorem

Suppose that

${\displaystyle g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots }$

is univalent in ${\displaystyle |z|>1}$. Then

${\displaystyle \sum _{n=1}^{\infty }n|b_{n}|^{2}\leq 1.}$

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

${\displaystyle \int _{X(r)}dx\,dy={1 \over 2i}\int _{\partial X(r)}{\overline {z}}\,dz={1 \over 2i}\int _{|z|=r}{\overline {g}}\,dg=\pi r^{2}-\pi \sum _{n=1}^{\infty }n|b_{n}|^{2}r^{-2n}.}$

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

This result was proved in 1914 by the Swedish mathematician Thomas Hakon Grönwall.

The Koebe function is defined by

${\displaystyle f(z)={\frac {z}{(1-z)^{2}}}=\sum _{n=1}^{\infty }nz^{n}}$

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

The rotated Koebe function is

${\displaystyle f_{\alpha }(z)={\frac {z}{(1-\alpha z)^{2}}}=\sum _{n=1}^{\infty }n\alpha ^{n-1}z^{n}}$

with ${\displaystyle \alpha }$ a complex number of absolute value ${\displaystyle 1}$. The Koebe function and its rotations are schlicht: that is, univalent (analytic and one-to-one) and satisfying ${\displaystyle f(0)=0}$ and ${\displaystyle f'(0)=1}$.

Let

${\displaystyle g(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots }$

be univalent in ${\displaystyle |z|<1}$. Then

${\displaystyle |a_{2}|\leq 2.}$

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

${\displaystyle g(z^{-2})^{-1/2}=z-{1 \over 2}a_{2}z^{-1}+\cdots .}$

Equality holds if and only if ${\displaystyle g}$ is a rotated Koebe function.

This result was proved by Ludwig Bieberbach in 1916 and provided the basis for his celebrated conjecture that ${\displaystyle |a_{n}|\leq n}$, proved in 1985 by Louis de Branges.

Applying an affine map, it can be assumed that

${\displaystyle f(0)=0,\,\,\,f^{\prime }(0)=1,}$

so that

${\displaystyle f(z)=z+a_{2}z^{2}+\cdots .}$

If ${\displaystyle w}$ is not in ${\displaystyle f(\mathbf {D} )}$, then

${\displaystyle h(z)={wf(z) \over w-f(z)}=z+(a_{2}+w^{-1})z^{2}+\cdots }$

is univalent in ${\displaystyle |z|<1}$.

Applying the coefficient inequality to ${\displaystyle f}$ and ${\displaystyle h}$ gives

${\displaystyle |w|^{-1}\leq |a_{2}|+|a_{2}+w^{-1}|\leq 4,}$

so that

${\displaystyle |w|\geq {1 \over 4}.}$

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.[1]

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

${\displaystyle {r \over (1+r)^{2}}\leq |f(z)|\leq {r \over (1-r)^{2}}}$

${\displaystyle {1-r \over (1+r)^{3}}\leq |f^{\prime }(z)|\leq {1+r \over (1-r)^{3}}}$

${\displaystyle {1-r \over 1+r}\leq \left|z{f^{\prime }(z) \over f(z)}\right|\leq {1+r \over 1-r}}$

with equality if and only if ${\displaystyle f}$ is a Koebe function

${\displaystyle f(z)={z \over (1-e^{i\theta }z)^{2}}.}$

• Bieberbach, Ludwig (1916), "Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln", S.-B. Preuss. Akad. Wiss.: 940–955
• Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, pp. 1–2, ISBN 0-387-97942-5
• Conway, John B. (1995), Functions of One Complex Variable II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94460-9
• Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, ISBN 0-387-90795-5
• Gronwall, T.H. (1914), "Some remarks on conformal representation", Annals of Mathematics, 16: 72–76, doi:10.2307/1968044
• Nehari, Zeev (1952), Conformal mapping, Dover, pp. 248–249, ISBN 0-486-61137-X
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht
• Rudin, Walter (1987). Real and Complex Analysis. Series in Higher Mathematics (3 ed.). McGraw-Hill. ISBN 0-07-054234-1. MR 0924157.

• Koebe 1/4 theorem at PlanetMath