Additional Problems for Chapter 6 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 6

  1. Find explicit Riemann maps for the slit disks $\DD \sm [0,1)$ and $\DD \sm [1/2,1)$. You may express your answers as compositions of explicit elementary conformal maps.
  2. Suppose $U \subset \CC$ is a bounded domain and $p \in U$. Let $\FF$ be the family of all injective holomorphic functions $f: \DD \to U$ with $f(0)=p$. Prove the following statements:

    (i) There is a $g \in \FF$ which satisfies $|g'(0)|=\sup_{f \in \FF} |f'(0)|$.

    (ii) If $V$ is any simply connected domain with $g(\DD) \subset V \subset U$, then $g(\DD)=V$.

  3. Let $c>0$. Show that the squaring map $z \mapsto z^2$ sends the family of ellipses with foci $\pm c$ to the family of ellipses with foci $0,c^2$. (Hint: Use the Zhukovskii map.)
  4. Suppose $f(z)=z+\sum_{n=2}^{\infty} a_n\, z^n \in {\mathcal S}$ and $f(\DD)$ is convex. Show that $|a_2|\leq 1$. (Hint: Apply the Schwarz lemma to the function $g: \DD \to \DD$ given by \[ g(z):=f^{-1} \left( \frac{f(\sqrt{z})+f(-\sqrt{z})}{2} \right), \] which is well-defined and holomorphic, with $g(0)=0$.)
  5. For each $q \neq 0$ find an injective holomorphic map $f: \DD \to \Chat$ with $f(0)=0, f'(0)=1$ such that $f(\DD)$ does not contain $q$. Explain why this is not in violation of Koebe's $1/4$-Theorem.
  6. Suppose $f$ is schlicht and $\DD(0,1/4)$ is the largest disk contained in $f(\DD)$, i.e., there is a $p \notin f(\DD)$ with $|p|=1/4$. Prove that $f$ is rotationally conjugate to the Koebe function $K$. As a corollary, it follows that $f(\DD)$ must be the slit-plane $\CC \sm \{ rp: r \geq 1 \}$.
  7. Show that normalized injective holomorphic functions on large disks are nearly affine: For every $R>0, \ve>0$ there is an $r_0=r_0(R,\ve)>0$ such that if $r>r_0$ and $f:\DD(0,r) \to \CC$ is injective and holomorphic with $f(0)=0,f'(0)=1$, then $|f(z)-z|<\ve$ for $|z|\leq R$.
  8. Suppose $f$ is schlicht and $f(\DD)$ is a bounded domain. Show that $(1-|z|)|f'(z)| \to 0$ uniformly as $|z| \to 1$. (Hint: Relate $(1-|z|)^2|f'(z)|^2$ to an area by Koebe's distortion bounds and use the fact that the area of $f(\{ z: 1-\ve<|z|<1)$ tends to $0$ as $\ve \to 0$.)

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