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\begin{document}

\centerline{\Large Riemann
  mappings and normal families}


\vskip 1cm


{\bf Riemann mappings.}

\begin{enumerate}[1.]
\item \label{1} Let $\Omega\subset\C$ be the domain pictured, a disk
  with a long and skinny rectangle attached at the bottom. Let
  $f:\Omega\to\mathbb D$ be the (inverse of the) Riemann map, and let
  ${\overline f}:{\overline\Omega}\to{\overline{\mathbb D}}$ be the
  homeomorphism extending $f$ (from the Carath\'eodory-Osgood theorem). Show
  that there is an arc $A\subset \partial\mathbb D$ whose endpoints
  are at distance $\leq 2\sqrt{\pi\epsilon}$ and so that $\overline f^{-1}(A)$
  contains the lower half of the boundary of the rectangle portion of
  $\Omega$. So if you traverse $\partial \mathbb D$ with constant speed, the
  image in $\partial\Omega$ under $\overline f^{-1}$ will speed up
  dramatically when it's in the lower half of the rectangle. Hint:
  Mimic the proof of Carath\'eodory-Osgood and show that the images of
  horizontal green arcs in the rectangle will on average have length
  $\leq\sqrt{\pi\epsilon}$.

  \begin{figure}[H]
    \centering
  \includegraphics[scale=0.5]{riemann.pdf}
  \end{figure}
 
\end{enumerate}



\subsection*{Maximum principle.}
       \vskip 0.5cm
       Should have assigned these a few weeks ago. Better late than never.
       \begin{enumerate}[resume]

\item Let $\Omega$ be a bounded domain and $f_n:\overline\Omega\to\C$
  continuous and
  holomorphic on $\Omega$. Show that
  if $f_n$ converge uniformly on $\partial\Omega$ then they converge
  uniformly on $\overline\Omega$.

\item Let $z_1,z_2,\cdots,z_n$ be on the unit circle $\{|z|=1\}$. Show
  that there is some $z\in\{|z|=1\}$ such that the product of the
  distances to all $z_k$ is $\geq 1$.

       \end{enumerate}

\subsection*{Normal families.}
\vskip 0.5cm


\begin{enumerate}[resume]
  \item Show that the family of functions $z\mapsto z^n$,
    $n=1,2,\cdots$ is normal on $\{|z|<1\}$ but not on any larger domain.
   \item Let $\mathcal F$ be a normal family of holomorphic functions
     $\Omega\to\C$ for a domain $\Omega$. Show that the family of
     derivatives $$\mathcal
     F'=\{f'\mid f\in\mathcal F\}$$ is also normal.
   \item Suppose $\mathcal F$ is a normal family of functions defined
     on $\mathbb D$. Show that there are constants $C_n$ so that if
     $f\in\mathcal F$ and $f(z)=\sum a_nz^n$ is the power series of
     $f$ around 0, then $|a_n|\leq C_n$ for all $n$.
  \item Let $\Omega\subset\C$ be a bounded domain (not necessarily
    simply connected) and $f:\Omega\to\Omega$ injective
    holomorphic. If $f(a)=a$ and $f'(a)=1$ for some $a\in\Omega$, show
    that $f$ is identity. Hint: Assuming $a=0$ and $f\neq id$ write
    $f(z)=z+bz^n+\cdots$ around 0. Argue that $\{f^k\mid
    k=1,2,\cdots\}$ is a normal family and write the beginning of a
    power series around 0 for $f^k$ to see that $b$ must be 0, using
    Problem 6. There is also a proof using the Schwarz lemma if we
    grant the Uniformization theorem, but the intended argument is
    with normal families.
  \item % for a solution of (i),(iii) see off_the_web
    Let $\Omega\subset\C$ be a domain. For a holomorphic function
  $f:\Omega\to\C$ define the $L^2$-norm of $f$ to be
  $$\norm {f}_{L^2(\Omega)}=\bigg(\int_{\Omega}|f|^2 d(\mbox{area})\bigg)^{1/2}$$
  \begin{enumerate}[(i)]
  \item For $\Omega=D(0,r)$ (disk centered at 0 and radius $r$) show that
    $$|f(0)|\leq \frac 1{r\sqrt\pi}\norm{f}_{L^2(D(0,r))}$$
    and deduce that for any $\Omega$ and any compact set
    $K\subset\Omega$ there is a constant $C_K$ so that
        $$\max_{z\in K}|f(z)|\leq C_K \norm {f}_{L_2}$$
        Thus the $L^{\infty}$-norm is bounded by (a multiple of) the
        $L^2$-norm.
      \item Show that
        $$\{f:\Omega\to\C\mid f\mbox{ is holomorphic and }\norm
        {f}_{L^2}\leq 1\}$$ is a normal family.
        \item Suppose $f$ is holomorphic on $\Omega=\mathbb
          D\smallsetminus\{0\}$. If $\norm{f}_{L^2(\Omega)}<\infty$
          show that 0 is a removable singularity.
  \end{enumerate}
  Hints and comments: This is a small variant of an old qual question
  and seems quite a bit harder than most qual questions. For (i) use
  the method as in Carath\'eodory-Osgood (or Problem 1 in this
  set). Start with $f(0)=\frac 1{2\pi}\int_0^{2\pi}f(\rho
  e^{i\theta})d\theta$. You'd like the area form $\rho ~d\rho d\theta$
  in the integral, so first multiply by $\rho$ and then integrate over
  $\rho\in (0,r)$. Then use Cauchy-Schwarz to replace $|f(z)| d(\mbox{area})$
  by $|f(z)|^2 d(\mbox{area})$. I don't know what normality has to do with
  (iii), but you can use the criterion that $0$ is removable if
  $\lim_{z\to 0} zf(z)=0$, and the estimate from (i) applied to
  smaller and smaller disks with 0 in the boundary (the $L^2$-norm on
  these disks goes to 0 by Lebesgue dominated convergence).

   
    
\end{enumerate}

\subsection*{Schwarz reflection}
\begin{enumerate}[resume]
  \item This is from McMullen's class at Harvard. Show that there is
    no proper holomorphic map $f:\D\to\C$. Hint: Extend continuously to
    the boundary and reflect in the boundary.

    \end{enumerate}


\end{document}
