Additional Problems for Chapter 12 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 12

  1. Suppose $U,V,W$ are finitely connected domains in $\Chat$ and $f:U \to V$ and $g:V \to W$ are proper holomorphic maps. Let $d_f \geq 1$ and $N_f \geq 0$ denote the mapping degree and number of critical points of $f$ counting multiplicities, and similarly define $d_g, N_g$. Show that $h=g \circ f:U \to W$ is proper and express $d_h, N_h$ in terms of the data for $f$ and $g$.
  2. Suppose $f \in \OO(\CC)$ is non-constant, $D$ is a disk and $U$ is a connected component of $f^{-1}(D)$. Show that $U$ is simply connected. Moreover, if $D$ contains an omitted value of $f$, show that $U$ must be unbounded.
  3. Prove the following version of the maximum principle: Suppose $U \subset \CC$ is an unbounded domain and $f \in \OO(U)$ extends continuously to $\ov{U}$. If $f$ is bounded in $U$ and $|f| \leq M$ on $\bd U$, then $|f| \leq M$ in $U$. The example of the exponential function in the right half-plane shows that the boundedness assumption in $U$ cannot be dispensed with. (Hint: First assume $\CC \sm U$ contains some disk $\DD(p,r)$. Take $z_0 \in U$ and $n \in \NN$, let $V$ be the connected component of $U \cap \DD(p,R)$ containing $z_0$, where $R>0$ is large. Apply the standard maximum principle to $(f(z))^n/(z-p)$ on $\ov{V}$ to get $|f(z_0)|\leq (|z_0-p|/r)^{1/n} M$, then let $n \to \infty$. For the general case, remove a small closed disk from $U$ and reduce to the previous case.)
  4. Show that if $q \in \CC$ is the omitted value of a non-constant $f \in \OO(\CC)$, then $q$ is an asymptotic value of $f$. (Hint: Use the results of the previous two problems to find a decreasing sequence $U_1 \supset U_2 \supset U_3 \supset \cdots$ of unbounded domains, where each $U_n$ is a connected component of $f^{-1}(\DD(q,1/n))$.)
  5. Prove that for every non-constant $f \in \OO(\CC)$ there is a curve $\gamma:[0,1) \to \CC$ such that $\gamma(t)$ and $f(\gamma(t))$ tend to $\infty$ as $t \to 1$.
  6. Let $P$ be a complex polynomial of degree $d \geq 1$. Show that there are at least $d+1$ distinct points $z \in \CC$ for which $P(z)(P(z)-1)=0$. (Hint: Use the Riemann-Hurwitz formula to find a lower bound for the number of distinct preimages of a set of $k$ points under $P$.)
  7. Suppose $f$ is a non-constant holomorphic function in the annulus $U=\{ z \in \CC: 1<|z|<2 \}$ such that $|f(z)| \to 1$ as $|z| \to 1$ or $|z|\to 2$. Show that, counting multiplicities, $f$ and $f'$ have the same number of zeros in $U$ which is $\geq 2$. (Hint: $f:U \to \DD$ is proper.)
  8. Let $f: \DD \to \DD$ be a proper holomorphic map (i.e., a finite Blaschke product). By Pick's theorem, the hyperbolic norm \[ \| f'(z) \| := \frac{\rho_{\DD}(f(z)) |f'(z)|}{\rho_{\DD}(z)} = \frac{(1-|z|^2)|f'(z)|}{1-|f(z)|^2} \] satisfies $\| f' \| \leq 1$, with $\| f' \|=1$ somewhere in $\DD$ if and only if $f \in \Aut(\DD)$, in which case $\| f' \|=1$ everywhere in $\DD$. By the chain rule the relations $\| (\varphi \circ f)' \|= \| f' \|$ and $\| (f\circ \varphi)' \|= \| f' \circ \varphi \|$ hold for every $\varphi\in \Aut(\DD)$. Show that $\| f'(z) \| \to 1$ as $|z| \to 1$. (Hint: It suffices to treat the case $z \to 1$, i.e., to show that $(1-|f(z)|^2)/(1-|z|^2) \to |f'(1)|$ as $z \to 1$. Use induction on the degree of $f$, noting that $zf'(z)/f(z)=|f'(z)|>0$ on the unit circle.)
  9. Suppose $f,g: \DD \to \DD$ are proper holomorphic maps having the same critical points of the same multiplicities. Prove that $f=\varphi \circ g$ for some $\varphi \in \Aut(\DD)$ by completing the following outline:

    (i) The ratio \[ u:=\frac{\|f'\|}{\|g'\|}=\frac{(\rho_{\DD} \circ f) |f'|}{(\rho_{\DD} \circ g) |g'|}, \] initially defined away from the critical points, extends to a positive smooth function in $\DD$, with $u(z) \to 1$ as $|z| \to 1$.

    (ii) $\Delta \log u = (\rho_{\DD} \circ f)^2 |f'|^2 - (\rho_{\DD} \circ g)^2 |g'|^2$ everywhere in $\DD$.

    (iii) $u \leq 1$ everywhere in $\DD$. If not, there is a point $p \in \DD$ where $u$ reaches its maximum $u(p)>1$. Use $\Delta \log u(p) \leq 0$ to arrive at a contradiction. By the same argument, $u \geq 1$. Hence $u=1$ everywhere in $\DD$.

    (iv) The condition $\|f'\|=\|g'\|$ everywhere in $\DD$ means that the assignment $g(z) \mapsto f(z)$ is a hyperbolic isometry as $z$ varies over any small region without critical points. Extending this isometry to some $\varphi \in \Aut(\DD)$, it follows that $f = \varphi \circ g$ in that region, hence everywhere.

  10. Suppose $f:\DD \to \DD$ is a proper holomorphic map of degree $3$. Prove that there exist $\varphi, \psi \in \Aut(\DD)$ such that $$ (\psi \circ f \circ \varphi)(z) = z \, \frac{z^2-a^2}{1-a^2z^2} \ \ \text{for all} \ z \in \DD. $$ Here the parameter $0 \leq a < 1$ is uniquely determined by $f$. (Hint: There is a disk automorphism that sends the two critical points of $f$ to a symmetric pair $\pm c$ for some $0 \leq c < 1$ uniquely determined by $f$. Check that the map $a \mapsto c(a)$ that assigns to the above Blaschke product its non-negative critical point is a homeomorphism $[0,1) \to [0,1)$.)

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