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We now approach conformal maps from yet another perspective. Given an open subset {U} of the complex numbers {{\bf C}}, define a univalent function on {U} to be a holomorphic function {f: U \rightarrow {\bf C}} that is also injective. We will primarily be studying this concept in the case when {U} is the unit disk {D(0,1) := \{ z \in {\bf C}: |z| < 1 \}}.

Clearly, a univalent function {f: D(0,1) \rightarrow {\bf C}} on the unit disk is a conformal map from {D(0,1)} to the image {f(D(0,1))}; in particular, {f(D(0,1))} is simply connected, and not all of {{\bf C}} (since otherwise the inverse map {f^{-1}: {\bf C} \rightarrow D(0,1)} would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset {V \subsetneq {\bf C}} of the complex numbers is the image of a univalent function on {D(0,1)}. Furthermore, if {V} contains the origin, then the univalent function {f: D(0,1) \rightarrow {\bf C}} with this image becomes unique once we normalise {f(0) = 0} and {f'(0) > 0}. Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions {f: D(0,1) \rightarrow {\bf C}} with {f(0)=0} and {f'(0)>0}. We will focus particular attention on the univalent functions {f: D(0,1) \rightarrow {\bf C}} with the normalisation {f(0)=0} and {f'(0)=1}; such functions will be called schlicht functions.

One basic example of a univalent function on {D(0,1)} is the Cayley transform {z \mapsto \frac{1+z}{1-z}}, which is a Möbius transformation from {D(0,1)} to the right half-plane {\{ \mathrm{Re}(z) > 0 \}}. (The slight variant {z \mapsto \frac{1-z}{1+z}} is also referred to as the Cayley transform, as is the closely related map {z \mapsto \frac{z-i}{z+i}}, which maps {D(0,1)} to the upper half-plane.) One can square this map to obtain a further univalent function {z \mapsto \left( \frac{1+z}{1-z} \right)^2}, which now maps {D(0,1)} to the complex numbers with the negative real axis {(-\infty,0]} removed. One can normalise this function to be schlicht to obtain the Koebe function

\displaystyle  f(z) := \frac{1}{4}\left( \left( \frac{1+z}{1-z} \right)^2 - 1\right) = \frac{z}{(1-z)^2}, \ \ \ \ \ (1)

which now maps {D(0,1)} to the complex numbers with the half-line {(-\infty,-1/4]} removed. A little more generally, for any {\theta \in {\bf R}} we have the rotated Koebe function

\displaystyle  f(z) := \frac{z}{(1 - e^{i\theta} z)^2} \ \ \ \ \ (2)

that is a schlicht function that maps {D(0,1)} to the complex numbers with the half-line {\{ -re^{-i\theta}: r \geq 1/4\}} removed.

Every schlicht function {f: D(0,1) \rightarrow {\bf C}} has a convergent Taylor expansion

\displaystyle  f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots

for some complex coefficients {a_1,a_2,\dots} with {a_1=1}. For instance, the Koebe function has the expansion

\displaystyle  f(z) = z + 2 z^2 + 3 z^3 + \dots = \sum_{n=1}^\infty n z^n

and similarly the rotated Koebe function has the expansion

\displaystyle  f(z) = z + 2 e^{i\theta} z^2 + 3 e^{2i\theta} z^3 + \dots = \sum_{n=1}^\infty n e^{(n-1)\theta} z^n.

Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients {a_n} should obey the bound {|a_n| \leq n} for all {n}. After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.

It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions {g: D(0,1) \rightarrow {\bf C}} that are odd, thus {g(-z)=-g(z)} for all {z}, and the Taylor expansion now reads

\displaystyle  g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots

for some complex coefficients {b_1,b_3,\dots} with {b_1=1}. One can transform a general schlicht function {f: D(0,1) \rightarrow {\bf C}} to an odd schlicht function {g: D(0,1) \rightarrow {\bf C}} by observing that the function {f(z^2)/z^2: D(0,1) \rightarrow {\bf C}}, after removing the singularity at zero, is a non-zero function that equals {1} at the origin, and thus (as {D(0,1)} is simply connected) has a unique holomorphic square root {(f(z^2)/z^2)^{1/2}} that also equals {1} at the origin. If one then sets

\displaystyle  g(z) := z (f(z^2)/z^2)^{1/2} \ \ \ \ \ (3)

it is not difficult to verify that {g} is an odd schlicht function which additionally obeys the equation

\displaystyle  f(z^2) = g(z)^2. \ \ \ \ \ (4)

Conversely, given an odd schlicht function {g}, the formula (4) uniquely determines a schlicht function {f}.

For instance, if {f} is the Koebe function (1), {g} becomes

\displaystyle  g(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \dots, \ \ \ \ \ (5)

which maps {D(0,1)} to the complex numbers with two slits {\{ \pm iy: y > 1/2 \}} removed, and if {f} is the rotated Koebe function (2), {g} becomes

\displaystyle  g(z) = \frac{z}{1- e^{i\theta} z^2} = z + e^{i\theta} z^3 + e^{2i\theta} z^5 + \dots. \ \ \ \ \ (6)

De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have

Theorem 1 (de Branges’ theorem) Let {n \geq 1} be a natural number.

  • (i) (Robertson conjecture) If {g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots} is an odd schlicht function, then

    \displaystyle  \sum_{k=1}^n |b_{2k-1}|^2 \leq n.

  • (ii) (Bieberbach conjecture) If {f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots} is a schlicht function, then

    \displaystyle  |a_n| \leq n.

It is easy to see that the Robertson conjecture for a given value of {n} implies the Bieberbach conjecture for the same value of {n}. Indeed, if {f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots} is schlicht, and {g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots} is the odd schlicht function given by (3), then from extracting the {z^{2n}} coefficient of (4) we obtain a formula

\displaystyle  a_n = \sum_{j=1}^n b_{2j-1} b_{2(n+1-j)-1}

for the coefficients of {f} in terms of the coefficients of {g}. Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of {n} from the Robertson conjecture for the same value of {n}. We remark that Littlewood and Paley had conjectured a stronger form {|b_{2k-1}| \leq 1} of Robertson’s conjecture, but this was disproved for {k=3} by Fekete and Szegö.

To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of {\log \frac{f(z)}{z}}, known as the Milin conjecture. Next, one continuously enlarges the image {f(D(0,1))} of the schlicht function to cover all of {{\bf C}}; done properly, this places the schlicht function {f} as the initial function {f = f_0} in a sequence {(f_t)_{t \geq 0}} of univalent maps {f_t: D(0,1) \rightarrow {\bf C}} known as a Loewner chain. The functions {f_t} obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term {\mu_t} (or {\theta(t)}, in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of {f(z)}, {g(z)}, or {\log \frac{f(z)}{z}}. After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of {n}) but for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.

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