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\begin{document}

\centerline{\Large Homework 7: Argument Principle, Rouch\'e,}
\vskip 0.3cm
  \centerline{\Large Maximum Modulus
  Principle}
\vskip 1cm

{\bf Argument Principle}

\begin{enumerate}[1.]

\item The curve A below is the image of $|z|=1$ under the map
  $z\mapsto (z+\frac 12)^2$. All pictures are drawn up to isotopy,
  i.e. deformations that don't change crossings. Can you determine the
  complementary component that contains 0? Prove that there are
  no holomorphic functions defined in $\{|z|<1.1\}$ that map $|z|=1$
  to any of the curves B,C,D. Orientations are induced by the
  counterclockwise orientation of $|z|=1$. Hint: For the curve D
  consider a double cover. Also note that $B$ is the image of $|z|=1$
  under $z\mapsto \frac 1z$, but that's not defined at $z=0$.
  \begin{center}
    \begin{figure}[H]
      \includegraphics[scale=0.5]{argument_principle.pdf}
    \end{figure}
  \end{center}

\item How many roots does $z^4+z+1=0$ have in the first quadrant?
  Hint: Consider the image under the map of the boundary of a big quarter disk.
\end{enumerate}

{\bf Rouch\'e's theorem.} %from Alpay
\begin{enumerate}[resume]
\item Show that all roots of $z^4+z^3+1=0$ have norm $<\frac 32$.

\item Prove the following special case of Brouwer's fixed point
  theorem (which states that any continuous map of a closed disk to
  itself has a fixed point). Let $f$ be holomorphic in $\{|z|<1.1\}$ and
  assume it maps $\{|z|\leq 1\}$ to $\{|z|<1\}$. Then $f$ has a fixed
  point.

\item How many roots does $z^4-3z+1=0$ have in $\{|z|<1\}$?

\item How many roots does $z^4+z^3-4z+1=0$ have in $\{1<|z|<3\}$?

\item How many roots does $e^z-4z^n+1=0$ have in $\{|z|<1\}$?


\end{enumerate}

{\bf Maximum Modulus Principle.} % from Ponnusamy-Silverman
\begin{enumerate}[resume]

\item Suppose $f$ is a holomorphic function in a neighborhood of
  $\{1\leq |z|\leq R\}$. Suppose that $|f(z)|\leq 1$ when $|z|\leq 1$
  and $|f(z)|\leq R$ when $|z|=R$. Prove that $|f(z)|\leq |z|$ when
  $1\leq |z|\leq R$.

\item Let $f$ be a nonconstant entire function and let
  $M_r=\max_{|z|=r} |f(z)|$. Show that $r\mapsto M_r$ is strictly
  increasing. 
  
\item Prove the Minimum Modulus Principle: If $\Omega\subseteq\C$ is a
  domain, $f:\Omega\to\C$ holomorphic and nonconstant, and if
  $f(z)\neq 0$ for $z\in \Omega$, then $|f|$ does not attain its
  infimum on $\Omega$.

\item Give an alternative proof (called the ``nobody is above
  average'' proof) of the Maximum Modulus Principle, as
  follows. Suppose $z_0\in\Omega$, $f:\Omega\to\C$ is holomorphic, and
  $|f(z_0)|\geq |f(z)|$ for $z\in\Omega$. Let $r>0$ such that
  $\{|z-z_0|\leq r\}\subset\Omega$. Recall the Mean Value Property of
  holomorphic functions
  $$f(z_0)=\frac 1{2\pi}\int_0^{2\pi} f(z_0+re^{it})~ dt$$
  Show that this implies that $|f(z)|$ is constant in a neighborhood
  of $z_0$ and then use Problem \#2 from Homework \#1.


\end{enumerate}

%{\bf Schwarz Lemma.}%Marshall
%\begin{enumerate}[resume]
%\item Let $f:\mathcal D\to\mathcal D$ be holomorphic where $\mathcal
%  D=\{|z|<1\}$. If $f(0)=a$ show that $f$ has no zeros in
%  $\{|z|<|a|\}$. 


%\end{enumerate}

  

\end{document}
