Additional Problems for Chapter 1 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 1

  1. Let $U \subset \CC$ be open. For a smooth function $g: U \to \RR$ the gradient vector field $\nabla g = (g_x,g_y)$ can be identified with the complex-valued function $g_x+ig_y$. Show that under this identification

    (i) $\nabla |f|^2 = 2 f \, \ov{f'}$ whenever $f \in \OO(U)$.

    (ii) $\nabla \log|f| = \dfrac{\ov{f'}}{\ov{f}}$ whenever $f \in \OO(U)$ is non-vanishing.

  2. Determine the radius of convergence of the power series \[ f(z) = \sum_{n=0}^{\infty} a_n \, z^n, \ \ \text{where} \ \ a_n = \sum_{k=0}^n \frac{1}{k!}. \] Then find a formula for $f(z)$ in terms of familiar elementary functions.
  3. Suppose $f$ is holomorphic in the disk ${\mathbb D}(0,R)$ and $M_n:=\sup_{|z| < R} |f^{(n)}(z)| < +\infty$. Show that for $|z| < R$, \[ \left| f(z)- \sum_{k=0}^{n-1} \frac{f^{(k)}(0)}{k!} z^k \right| \leq \frac{M_n |z|^n}{n!}. \] (Hint: Use the fact that the remainder $f(z)- \sum_{k=0}^{n-1} (f^{(k)}(0)/k!) z^k$ has a zero of order $\geq n$ at $z=0$, and its $n$-th derivative is $f^{(n)}$ everywhere in $\DD(0,R)$.)
  4. Suppose $f$ is holomorphic in the disk ${\mathbb D}(0,R)$ and $M_0:=\sup_{|z| < R} |f(z)| < +\infty$. Show that for $|z| < R$, \[ \left| f(z)- \! \sum_{k=0}^{n-1} \frac{f^{(k)}(0)}{k!} z^k \right| \! \leq \! \frac{M_0 |z|^n}{R^{n-1}(R-|z|)}. \] (Hint: Use Cauchy's estimates (1.21) to majorize $|f^{(k)}(0)|$ for $k \geq n$.)
  5. Suppose the polynomial $P(z)=a_0+a_1 z+ \cdots + a_n z^n$ satisfies $|P(z)| \leq 1$ whenever $|z|=1$. Verify the following:

    (i) $|a_k|\leq 1$ for all $0 \leq k \leq n$.

    (ii) $|P(z)| \leq |z|^n$ whenever $|z|>1$.

    (iii) If $|P(z)|=1$ whenever $|z|=1$, then $P(z)=a_k z^k$ for some $0 \leq k \leq n$ with $|a_k|=1$.

  6. (Hörmander) Suppose $f$ is holomorphic in a neighborhood of $\ov{\DD}$ and $P$ is a monic polynomial. Show that \[ |f(0)| \leq \sup_{|z|=1} |f(z)P(z)|. \]
  7. The case of equality in Cauchy's estimates: Suppose $f$ is continuous on $\ov{\DD}(p,r)$ and holomorphic in $\DD(p,r)$. If \[ |f^{(n)}(p)| = \frac{n!}{r^n} \ \sup_{|z-p|=r} |f(z)| \] for some $n\geq 0$, show that $f(z)=cz^n$ for some $c \in \CC$. (Hint: Parseval's formula can be useful.)
  8. Suppose $f$ is continuous on $\ov{\DD}$ and holomorphic in $\DD$. If $f(z)=f(1/z)$ for all $z \in \TT$, show that $f$ is constant.
  9. Suppose $f \in \OO(\CC)$ satisfies $|f(z)| \leq |\myre(z)|^{-1/2}$ for all $z$ off the imaginary axis. Prove that $f \equiv 0$. (Hint: Write Cauchy's integral formula for $f^{(n)}(0)$ where the path of integration is the boundary of a large square centered at $0$.)
  10. Suppose $f=u+iv$ is an entire function such that $uv: \CC \to \RR$ is bounded. Show that $f$ is constant.
  11. Suppose $f,g$ are holomorphic in a domain $U$ and there is an $\alpha>0$ such that $|f|^\alpha+|g|^\alpha=1$ in $U$. Show that $f,g$ are constant in $U$.
  12. Suppose $f: U \times U \to \CC$ is continuous and holomorphic in each variable (i.e., for each $p \in U$ the functions $z \mapsto f(z,p)$ and $z \mapsto f(p,z)$ are holomorphic in $U$). Show that $g(z):=f(z,z)$ is holomorphic in $U$. (Hint: One possible idea is to use Theorem 1.46. A result of F. Hartogs shows that the assumption of continuity is in fact redundant.)
  13. Let $U \subset \CC$ be open and connected. What can you say about a non-constant holomorphic function $f: U \to U$ which satisfies $f \circ f = f$?
  14. Show that there is a holomorphic function $f$ defined near the origin which satisfies $f(0)=0$ and $f(z) \cos(f(z)) =z$ for all $z$ close to $0$. Find the first three non-zero terms in the power series expansion of $f$ about $0$.
  15. (C. L. Petersen) Suppose $f \in \OO(\DD)$ and $f(0) \in \Dstar$. If $\myre(f'(z))>0$ for every $z$ on the segment $[0,f(0)]$, show that $[0,f(0)]$ does not contain any fixed point of $f$. (Hint: For $z \in [0,f(0)]$, write $f(z)-z=f(0)+\int_{[0,z]} (f'(\zeta)-1) \, d\zeta$.)
  16. Let $U$ be the open unit square $\{ z \in \CC: |\myre(z)| < 1, |\myim(z)| < 1 \}$. Suppose $f$ is continuous on $\ov{U}$ and holomorphic in $U$. If $A,B,C,D$ denote the maximum values of $|f|$ on the four sides of $\bd U$, show that $|f(0)|\leq (ABCD)^{1/4}$.
  17. Let $P: \CC \to \CC$ be a polynomial of degree $d$ with integer coefficients such that $\sup_{|z|=1} |P(z)| \leq \sqrt{2}$. Show that $P(z)=\pm z^d$. (Hint: Use Parseval's formula.)

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