Additional Problems for Chapter 11 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 11

  1. Describe all holomorphic functions $f:\CC \sm \{ 0 \} \to \CC \sm \{ 0, 1 \}$.
  2. Let $f$ be a non-constant entire function which satisfies $f \circ \varphi = \varphi \circ f$ for some non-identity $\varphi \in \Aut(\CC)$. Show that $f(\CC)=\CC$.
  3. Prove that the equation $\sin z = z^2$ has infinitely many solutions in $\CC$.
  4. Prove that for every $q \in \CC$ the equation $e^{2z}+e^{3z}=q$ has infinitely many solutions in $\CC$.
  5. Show that the sequence $\{ f_n \}$ of entire functions defined by $f_1(z)=\sin(z)$ and $f_{n+1}(z)=\sin(f_n(z))$ for $n \geq 1$ is not normal in any neighborhood of the origin.
  6. Fix a domain $U \subset \CC$ and an integer $k \geq 1$. Let $\FF \subset \OO(U)$ be the family of all non-vanishing functions which take the value $1$ at most $k$ times. Show that $\FF$ is a normal family.
  7. Let $U \subset \CC$ be a domain and $\FF$ be the family of all $f \in \OO(U)$ for which $|f'(z)| \leq \exp(|f(z)|)$ for all $z \in U$. Use Zalcman's theorem to show that $\FF$ is a normal family.
  8. Here is a generalization of the previous exercise, due to Royden: Suppose a family $\FF \subset \MM(U)$ has the property that for each compact set $K \subset U$ there is an increasing function $h_K: [0,+\infty) \to [0,+\infty)$ such that $|f'(z)| \leq h_K(|f(z)|)$ for all $f \in \FF$ and all $z \in K$. Use Zalcman's theorem to prove that $\FF$ is a normal family. Is the assumption of $h_K$ being increasing essential for your proof? How do you reconcile this result with Marty's necessary and sufficient condition for normality?
  9. Let $f \in \OO(U)$ be nowhere-vanishing and $F \in \OO(U)$ be a primitive of $f$. Show that $g =|f(z)| \, |dz|$ is a smooth conformal metric in $U$ which is flat in the sense that $K_g=0$ everywhere. Show that $w=F(z)$ is a local isometry between $(U,g)$ and $(F(U),|dw|)$.

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