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\title{Math 336 Midterm, May 23, 2008}
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Name:\hrulefill

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One notebook sized page of notes is allowed.
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\item  Let $u$ and $v$ be harmonic on an open connected set $W$.  Suppose
  that $u(z)v(z)=0$ on an open subset of $W$.  Prove that either $u$ or
  $v$ is identically $0$ on $W$.



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\item  Using Rouch\'e's theorem determine the number of roots of the
  equation
$\ds z^7-4z^3+z+1=0$ inside the disk $\ds \{z: |z|<1\}$.

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\item Let $f$ be analytic on $\ds D=\{|z|<1\}$.  Suppose $\ds
  f(0)=f'(0)=0$ and $\ds |f(z)|<M$ for $\ds z\in D$.  Prove that $\ds
  |f(z)|\le M|z|^2$ for $z\in D$. 



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\item  Let $\Omega$ be a bounded connected open set.  Suppose $\ds 0\in
  \Omega$ and that $f$ is an analytic function on $\Omega$ such that if
  $\ds z\in 
  \Omega,\ f(z)\in \Omega$.  Suppose also that $\ds f(0)=0,\ f'(0)=1$.
  Prove that $\ds f(z)=z$.  Hint: use Cauchy's inequalities on the
  functions obtained by composing $f$ with itself $k$ times.


 

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