Additional Problems for Chapter 8 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 8

  1. Construct an entire function which has a zero of order $2$ at every point of the sequence $\{ \log n \}_{n \in \NN}$ and does not vanish anywhere else.
  2. Let $f \in \MM(\CC)$. Show that $f=g'/g$ for some $g \in \MM(\CC)$ if and only if the poles of $f$ are simple with integer residues. (Hint: For the “if” part, construct an $h \in \MM(\CC)$ with a zero or pole of order $\pm \res(f,p)$ at every pole $p$ of $f$, and with no other zeros or poles.)
  3. The Mahler measure $M(P)$ of a complex polynomial map $P$ is defined as the geometric mean of $|P|$ on the unit circle: \[ M(P) := \exp \left( \frac{1}{2\pi} \int_0^{2\pi} \log|P(e^{it})| \, dt \right). \] If $P(z)=C \prod_{n=1}^d (z-z_n)$, show that \[ M(P) = |C| \prod_{n=1}^d \max \{|z_n|,1\}. \]
  4. Let $a_n=1-1/n^2$. Prove that the infinite product \[ f(z)=\prod_{n=1}^{\infty} \left( \frac{a_n-z}{1-a_n z} \right) \] defines a holomorphic function $f: \DD \to \DD$.
  5. Determine all bounded holomorphic functions $f:\HH \to \CC$ such that $f(in)=e^{-n}$ for all $n \in \NN$.
  6. Suppose $f \in \OO(\CC)$ satisfies $|f(z)| \leq M e^{|z|}$ for all $z$. Show that \[ |f^{(n)}(0)| \leq M n! \left( \frac{e}{n} \right)^n \qquad n \geq 1. \] Conclude that $|f^{(n)}(0)|/\sqrt{n}$ stays bounded as $n \to \infty$.
  7. Let $f$ be a transcendental entire function with finitely many zeros. Show that $m(r):=\min_{|z|=r} |f(z)| \to 0$ as $r \to +\infty$.

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