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Grönwall's area theorem

Main article: Area theorem (conformal mapping)

Suppose that

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

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

∑ n = 1 ∞ n | b n | 2 ≤ 1. {\displaystyle \sum _{n=1}^{\infty }n|b_{n}|^{2}\leq 1.}

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

∫ X ( r ) d x d y = 1 2 i ∫ ∂ X ( r ) z ¯ d z = 1 2 i ∫ | z | = r g ¯ d g = π r 2 − π ∑ n = 1 ∞ n | b n | 2 r − 2 n . {\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 r {\displaystyle r} decrease to 1 {\displaystyle 1} . The above proof shows equality holds if and only if the complement of the image of g {\displaystyle g} has zero area, i.e. Lebesgue measure zero.

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

Koebe function

The Koebe function is defined by

f ( z ) = z ( 1 − z ) 2 = ∑ n = 1 ∞ n z n {\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 1 / 4 {\displaystyle 1/4} in the theorem cannot be improved, as the image domain f ( D ) {\displaystyle f(\mathbf {D} )} does not contain the point z = − 1 / 4 {\displaystyle z=-1/4} and so cannot contain any disk centred at 0 {\displaystyle 0} with radius larger than 1 / 4 {\displaystyle 1/4} .

The rotated Koebe function is

f α ( z ) = z ( 1 − α z ) 2 = ∑ n = 1 ∞ n α n − 1 z n {\displaystyle f_{\alpha }(z)={\frac {z}{(1-\alpha z)^{2}}}=\sum _{n=1}^{\infty }n\alpha ^{n-1}z^{n}}

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

Bieberbach's coefficient inequality for univalent functions

Let

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

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

| a 2 | ≤ 2. {\displaystyle |a_{2}|\leq 2.}

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

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

Equality holds if and only if g {\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 | a n | ≤ n {\displaystyle |a_{n}|\leq n} , proved in 1985 by Louis de Branges.

Proof of quarter theorem

Applying an affine map, it can be assumed that

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

so that

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

In particular, the coefficient inequality gives that | a 2 | ≤ 2 {\displaystyle |a_{2}|\leq 2} . If w {\displaystyle w} is not in f ( D ) {\displaystyle f(\mathbf {D} )} , then

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

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

Applying the coefficient inequality to h {\displaystyle h} gives

| w | − 1 = | w − 1 | = | − a 2 + a 2 + w − 1 | ≤ | a 2 | + | a 2 + w − 1 | ≤ 4 , {\displaystyle |w|^{-1}=|w^{-1}|=|-a_{2}+a_{2}+w^{-1}|\leq |a_{2}|+|a_{2}+w^{-1}|\leq 4,}

so that

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

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.¹

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

r ( 1 + r ) 2 ≤ | f ( z ) | ≤ r ( 1 − r ) 2 {\displaystyle {r \over (1+r)^{2}}\leq |f(z)|\leq {r \over (1-r)^{2}}} 1 − r ( 1 + r ) 3 ≤ | f ′ ( z ) | ≤ 1 + r ( 1 − r ) 3 {\displaystyle {1-r \over (1+r)^{3}}\leq |f^{\prime }(z)|\leq {1+r \over (1-r)^{3}}} 1 − r 1 + r ≤ | z f ′ ( z ) f ( z ) | ≤ 1 + r 1 − r {\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 f {\displaystyle f} is a Koebe function

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

Notes

• 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 (1/4): 72–76, doi:10.2307/1968044, JSTOR 1968044
• Nehari, Zeev (1952), Conformal mapping, Dover, pp. 248–249, ISBN 0-486-61137-X {{citation}}: ISBN / Date incompatibility (help)
• 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.

External links

• Koebe 1/4 theorem at PlanetMath

References

1. Pommerenke 1975, pp. 21–22 - Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht ↩