Additional Problems for Chapter 5 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 5

  1. Give an example of a sequence $\{ f_n \}$ of rational functions such that $f_n \to 1$ compactly in $\DD$ and $f_n \to 0$ compactly in $\CC \sm \ov{\DD}$.
  2. Suppose $U \subset \CC$ is a domain and the sequence $f_n \in \OO(U)$ converges compactly to $f \in \OO(U)$, where $f$ is not identically $0$. If $f(p)=0$ for some $p \in U$, show that for all large $n$ there is a zero $z_n$ of $f_n$ such that $z_n \to p$ as $n \to \infty$.
  3. Show that the series $\sum_{n=1}^{\infty} (-1)^n/(z-n)$ converges compactly in $\CC \sm \NN$. (Hint: Evidently the Weierstrass $M$-test won't do the trick. Instead, show that the sequence of partial sums satisfies a uniform Cauchy condition on each compact subset of $\CC \sm \NN$.)
  4. Suppose $f_n \in \OO(U)$ and $f_n \to f$ compactly in $U$. Let $p \in U$ and $\{ z_n \}$ be any sequence in $U \sm \{ p \}$ that converges to $p$. Show that \[ \lim_{n \to \infty} \frac{f_n(z_n)-f_n(p)}{z_n-p}=f'(p). \]
  5. Let $f: \DD \to \DD$ be a function with the property that for any triple $z_1, z_2, z_3$ of distinct points in $\DD$ there is a holomorphic function $g: \DD \to \DD$ (depending on $z_1,z_2,z_3$) with $g(z_i)=f(z_i)$ for $i=1,2,3$. Show that $f$ is holomorphic in $\DD$. (Hint: For any $p \in \DD$ and any two sequences $\{ z_n \}, \{ w_n \}$ in $\DD \sm \{ p \}$ converging to $p$, show that the sequences $(f(z_n)-f(p))/(z_n-p)$ and $(f(w_n)-f(p))/(w_n-p)$ have subsequences which converge to a common limit.)
  6. Let $\psi:(0,1) \to (0,+\infty)$ be any function. Show that the family of $f \in \OO(\DD)$ for which \[ \frac{1}{2\pi} \int_0^{2\pi} |f(re^{it})| \, dt \leq \psi(r) \] for all $0 < r < 1$ is precompact.
  7. Suppose $f_n: \DD \to \DD$ is holomorphic and $f_n(p) \to 1$ for some $p \in \DD$. Show that $f_n \to 1$ compactly in $\DD$.
  8. Let $M>0$ and $\FF$ be the family of all $f \in \OO(\DD)$ such that $f(0)=0$, $f'(0)=1$ and $|f|\leq M$. Show that there is a constant $\beta>0$ such that $f(\DD) \supset \DD(0,\beta)$ for all $f \in \FF$.
  9. Suppose $U \subset \CC$ is a domain, $f_n \in \OO(U)$, and $f(z)=\lim_{n \to \infty} f_n(z)$ exists for every $z \in U$. If $|f_n-1| \geq 1$ for all $n$, show that $f_n \to f$ compactly in $U$.

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