Loewner differential equation

Let ${\displaystyle f}$ and ${\displaystyle g}$ be holomorphic univalent functions on the unit disk ${\displaystyle D}$, ${\displaystyle |z|<1}$, with ${\displaystyle f(0)=0=g(0)}$.

${\displaystyle f}$ is said to be subordinate to ${\displaystyle g}$ if and only if there is a univalent mapping ${\displaystyle \varphi }$ of ${\displaystyle D}$ into itself fixing ${\displaystyle 0}$ such that

${\displaystyle \displaystyle {f(z)=g(\varphi (z))}}$

for ${\displaystyle |z|<1}$.

A necessary and sufficient condition for the existence of such a mapping ${\displaystyle \varphi }$ is that

${\displaystyle f(D)\subseteq g(D).}$

Necessity is immediate.

Conversely ${\displaystyle \varphi }$ must be defined by

${\displaystyle \displaystyle {\varphi (z)=g^{-1}(f(z)).}}$

By definition φ is a univalent holomorphic self-mapping of ${\displaystyle D}$ with ${\displaystyle \varphi (0)=0}$.

Since such a map satisfies ${\displaystyle 0<|\varphi '(0)|\leq 1}$ and takes each disk ${\displaystyle D_{r}}$, ${\displaystyle |z|<r}$ with ${\displaystyle 0<r<1}$, into itself, it follows that

${\displaystyle \displaystyle {|f^{\prime }(0)|\leq |g^{\prime }(0)|}}$

and

${\displaystyle \displaystyle {f(D_{r})\subseteq g(D_{r}).}}$

For ${\displaystyle 0\leq t\leq \infty }$ let ${\displaystyle U(t)}$ be a family of open connected and simply connected subsets of ${\displaystyle \mathbb {C} }$ containing ${\displaystyle 0}$, such that

${\displaystyle U(s)\subsetneq U(t)}$

if ${\displaystyle s<t}$,

${\displaystyle U(t)=\bigcup _{s<t}U(s)}$

and

${\displaystyle U(\infty )=\mathbb {C} .}$

Thus if ${\displaystyle s_{n}\uparrow t}$,

${\displaystyle U(s_{n})\rightarrow U(t)}$

in the sense of the Carathéodory kernel theorem.

If ${\displaystyle D}$ denotes the unit disk in ${\displaystyle \mathbb {C} }$, this theorem implies that the unique univalent maps ${\displaystyle f_{t}(z)}$

${\displaystyle f_{t}(D)=U(t),\,\,\,f_{t}(0)=0,\,\,\,\partial _{z}f_{t}(0)=1}$

given by the Riemann mapping theorem are uniformly continuous on compact subsets of ${\displaystyle [0,\infty )\times D}$.

Moreover, the function ${\displaystyle a(t)=f_{t}^{\prime }(0)}$ is positive, continuous, strictly increasing and continuous.

By a reparametrization it can be assumed that

${\displaystyle f_{t}^{\prime }(0)=e^{t}.}$

Hence

${\displaystyle f_{t}(z)=e^{t}z+a_{2}(t)z^{2}+\cdots }$

The univalent mappings ${\displaystyle f_{t}(z)}$ are called a Loewner chain.

The Koebe distortion theorem shows that knowledge of the chain is equivalent to the properties of the open sets ${\displaystyle U(t)}$.

If ${\displaystyle f_{t}(z)}$ is a Loewner chain, then

${\displaystyle \displaystyle {f_{s}(D)\subsetneq f_{t}(D)}}$

for ${\displaystyle s<t}$ so that there is a unique univalent self mapping of the disk ${\displaystyle \varphi _{s,t}(z)}$ fixing ${\displaystyle 0}$ such that

${\displaystyle \displaystyle {f_{s}(z)=f_{t}(\varphi _{s,t}(z)).}}$

By uniqueness the mappings ${\displaystyle \varphi _{s,t}}$ have the following semigroup property:

${\displaystyle \displaystyle {\varphi _{t,r}\circ \varphi _{s,t}=\varphi _{s,r}}}$

for ${\displaystyle s\leq t\leq r}$.

They constitute a Loewner semigroup.

The self-mappings depend continuously on ${\displaystyle s}$ and ${\displaystyle t}$ and satisfy

${\displaystyle \displaystyle {\varphi _{t,t}(z)=z.}}$

The Loewner differential equation can be derived either for the Loewner semigroup or equivalently for the Loewner chain.

For the semigroup, let

${\displaystyle \displaystyle {w_{s}(z)=\partial _{t}\varphi _{s,t}(z)|_{t=s}}}$

then

${\displaystyle \displaystyle {w_{s}(z)=-zp_{s}(z)}}$

with

${\displaystyle \displaystyle {\Re \,p_{s}(z)>0}}$

for ${\displaystyle |z|<1}$.

Then ${\displaystyle w(t)=\varphi _{s,t}(z)}$ satisfies the ordinary differential equation

${\displaystyle \displaystyle {{dw \over dt}=-wp_{t}(w)}}$

with initial condition ${\displaystyle w(s)=z}$.

To obtain the differential equation satisfied by the Loewner chain ${\displaystyle f_{t}(z)}$ note that

${\displaystyle \displaystyle {f_{t}(z)=f_{s}(\varphi _{s,t}(z))}}$

so that ${\displaystyle f_{t}(z)}$ satisfies the differential equation

${\displaystyle \displaystyle {\partial _{t}f_{t}(z)=zp_{t}(z)\partial _{z}f_{t}(z)}}$

with initial condition

${\displaystyle \displaystyle {f_{t}(z)|_{t=0}=f_{0}(z).}}$

The Picard–Lindelöf theorem for ordinary differential equations guarantees that these equations can be solved and that the solutions are holomorphic in ${\displaystyle z}$.

The Loewner chain can be recovered from the Loewner semigroup by passing to the limit:

${\displaystyle \displaystyle {f_{s}(z)=\lim _{t\rightarrow \infty }e^{t}\phi _{s,t}(z).}}$

Finally given any univalent self-mapping ${\displaystyle \psi (z)}$ of ${\displaystyle D}$, fixing ${\displaystyle 0}$, it is possible to construct a Loewner semigroup ${\displaystyle \varphi _{s,t}(z)}$ such that

${\displaystyle \displaystyle {\varphi _{0,1}(z)=\psi (z).}}$

Similarly given a univalent function ${\displaystyle g}$ on ${\displaystyle D}$ with ${\displaystyle g(0)=0}$, such that ${\displaystyle g(D)}$ contains the closed unit disk, there is a Loewner chain ${\displaystyle f_{t}(z)}$ such that

${\displaystyle \displaystyle {f_{0}(z)=z,\,\,\,f_{1}(z)=g(z).}}$

Results of this type are immediate if ${\displaystyle \psi }$ or ${\displaystyle g}$ extend continuously to ${\displaystyle \partial D}$. They follow in general by replacing mappings ${\displaystyle f(z)}$ by approximations ${\displaystyle f(rz)/r}$ and then using a standard compactness argument.[1]

Holomorphic functions ${\displaystyle p(z)}$ on ${\displaystyle D}$ with positive real part and normalized so that ${\displaystyle p(0)=1}$ are described by the Herglotz representation theorem:

${\displaystyle \displaystyle {p(z)=\int _{0}^{2\pi }{1+e^{-i\theta }z \over 1-e^{-i\theta }z}\,d\mu (\theta ),}}$

where ${\displaystyle \mu }$ is a probability measure on the circle. Taking a point measure singles out functions

${\displaystyle \displaystyle {p_{t}(z)={1+\kappa (t)z \over 1-\kappa (t)z}}}$

with ${\displaystyle |\kappa (t)|=1}$, which were the first to be considered by Loewner (1923).

Inequalities for univalent functions on the unit disk can be proved by using the density for uniform convergence on compact subsets of slit mappings. These are conformal maps of the unit disk onto the complex plane with a Jordan arc connecting a finite point to ∞ omitted. Density follows by applying the Carathéodory kernel theorem. In fact any univalent function ${\displaystyle f(z)}$ is approximated by functions

${\displaystyle \displaystyle {g(z)=f(rz)/r}}$

which take the unit circle onto an analytic curve. A point on that curve can be connected to infinity by a Jordan arc. The regions obtained by omitting a small segment of the analytic curve to one side of the chosen point converge to ${\displaystyle g(D)}$ so the corresponding univalent maps of ${\displaystyle D}$ onto these regions converge to ${\displaystyle g}$ uniformly on compact sets.[2]

To apply the Loewner differential equation to a slit function ${\displaystyle f}$, the omitted Jordan arc ${\displaystyle c(t)}$ from a finite point to ${\displaystyle \infty }$ can be parametrized by ${\displaystyle [0,\infty )}$ so that the map univalent map ${\displaystyle f_{t}}$ of ${\displaystyle D}$ onto ${\displaystyle \mathbb {C} }$ less ${\displaystyle c([t,\infty ))}$ has the form

${\displaystyle \displaystyle {f_{t}(z)=e^{t}(z+b_{2}(t)z^{2}+b_{3}(t)z^{3}+\cdots )}}$

with ${\displaystyle b_{n}}$ continuous. In particular

${\displaystyle \displaystyle {f_{0}(z)=f(z).}}$

For ${\displaystyle s\leq t}$, let

${\displaystyle \displaystyle {\varphi _{s,t}(z)=f_{t}^{-1}\circ f_{s}(z)=e^{s-t}(z+a_{2}(s,t)z^{2}+a_{3}(s,t)z^{3}+\cdots )}}$

with ${\displaystyle a_{n}}$ continuous.

This gives a Loewner chain and Loewner semigroup with

${\displaystyle \displaystyle {p_{t}(z)={1+\kappa (t)z \over 1-\kappa (t)z}}}$

where ${\displaystyle \kappa }$ is a continuous map from ${\displaystyle [0,\infty )}$ to the unit circle.[3]

To determine ${\displaystyle \kappa }$, note that ${\displaystyle \varphi _{s,t}}$ maps the unit disk into the unit disk with a Jordan arc from an interior point to the boundary removed. The point where it touches the boundary is independent of ${\displaystyle s}$ and defines a continuous function ${\displaystyle \lambda (t)}$ from ${\displaystyle [0,\infty )}$ to the unit circle. ${\displaystyle \kappa (t)}$ is the complex conjugate (or inverse) of ${\displaystyle \lambda (t)}$:

${\displaystyle \displaystyle {\kappa (t)=\lambda (t)^{-1}.}}$

Equivalently, by Carathéodory's theorem ${\displaystyle f_{t}}$ admits a continuous extension to the closed unit disk and ${\displaystyle \lambda (t)}$, sometimes called the driving function, is specified by

${\displaystyle \displaystyle {f_{t}(\lambda (t))=c(t).}}$

Not every continuous function ${\displaystyle \kappa }$ comes from a slit mapping, but Kufarev showed this was true when ${\displaystyle \kappa }$ has a continuous derivative.

Loewner (1923) used his differential equation for slit mappings to prove the Bieberbach conjecture

${\displaystyle \displaystyle {|a_{3}|\leq 3}}$

for the third coefficient of a univalent function

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

In this case, rotating if necessary, it can be assumed that ${\displaystyle a_{3}}$ is non-negative.

Then

${\displaystyle \displaystyle {\varphi _{0,t}(z)=e^{-t}(z+a_{2}(t)z^{2}+a_{3}(t)z^{3}+\cdots )}}$

with ${\displaystyle a_{n}}$ continuous. They satisfy

${\displaystyle \displaystyle {a_{n}(0)=0,\,\,a_{n}(\infty )=a_{n}.}}$

If

${\displaystyle \displaystyle {\alpha (t)=e^{-t}\kappa (t),}}$

the Loewner differential equation implies

${\displaystyle \displaystyle {{\dot {a}}_{2}=-2\alpha }}$

and

${\displaystyle \displaystyle {{\dot {a}}_{3}=-2\alpha ^{2}-4\alpha \,a_{2}.}}$

So

${\displaystyle \displaystyle {a_{2}=-2\int _{0}^{\infty }\alpha (t)\,dt}}$

which immediately implies Bieberbach's inequality

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

Similarly

${\displaystyle \displaystyle {a_{3}=-2\int _{0}^{\infty }\alpha (t)^{2}\,dt+4\left(\int _{0}^{\infty }\alpha (t)\,dt\right)^{2}}}$

Since ${\displaystyle a_{3}}$ is non-negative and ${\displaystyle |\kappa (t)|=1}$,

${\displaystyle \displaystyle {|a_{3}|=2\int _{0}^{\infty }|\Re \alpha (t)^{2}|\,dt+4\left(\int _{0}^{\infty }\Re \alpha (t)\,dt\right)^{2}}\leq 2\int _{0}^{\infty }|\Re \alpha (t)^{2}|\,dt+4\left(\int _{0}^{\infty }e^{-t}\,dt\right)\left(\int _{0}^{\infty }e^{t}(\Re \alpha (t))^{2}\,dt\right)=1+4\int _{0}^{\infty }(e^{-t}-e^{-2t})(\Re \kappa (t))^{2}\,dt\leq 3,}$

using the Cauchy–Schwarz inequality.

• Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, ISBN 0-387-90795-5
• Kufarev, P. P. (1943), "On one-parameter families of analytic functions", Mat. Sbornik, 13: 87–118
• Lawler, G. F. (2005), Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, ISBN 0-8218-3677-3
• Loewner, C. (1923), "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I", Math. Ann., 89: 103–121, doi:10.1007/BF01448091, hdl:10338.dmlcz/125927
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht