\magnification=1200
\baselineskip=15pt
\nopagenumbers
\centerline{\bf Complex Prelim, January 2006}
\bigskip
\item{1.} Suppose
$$p(z)=\sum_{n=0}^Na_nz^n$$
is a complex polynomial. Show that
$${1\over2\pi}\int_0^{2\pi}|p(e^{i\theta})|^2\,d\theta=\sum_{n=0}^N|a_n|^2.$$
\item{2.} If $u$ is a harmonic function defined on the complex plane and $f$ is entire,
show that $u\circ f$ is harmonic.
\medskip
\item{3.} Construct a conformal mapping from the first quadrant of the complex
plane onto the horizontal strip $|y|<1$.
\medskip
\item{4.} If $f(z)$ is an entire function and its real part is bounded from below,
show that $f$ must be constant.
\medskip
\item{5.} Find the Laurent expansion of the function
$$f(z)={2\over z(z-1)(z-2)}$$
in the annulus $1<|z|<2$.
\medskip
\item{6.} Suppose $f(z)$ is entire and $p(z)$ is a polynomial. If $|f(z)|\le|p(z)|$
for all $z$, show that there exists a constant $c$ such that $f(z)=cp(z)$.
\medskip
\item{7.} Characterize all anaytic functions $f(z)$ in $|z|<1$ such that
$|f(z)|\le|\sin(1/z)|$ for all $0<|z|<1$.
\medskip
\item{8.} Suppose each $f_n(z)$ is analytic in the unit disk $|z|<1$. If
$\sum|f_n(z)|$ converges uniformly for $|z|<1$, show that $\sum|f_n'(z)|$
converges uniformly for $|z|\le r$, where $r\in(0,1)$.
\medskip
\item{9.} If $f(z)$ is analytic in $|z|<1$ and $f'(0)\not=0$, prove the existence
of an analytic function $g(z)$ such that $f(z^n)=f(0)+g(z)^n$ in a neighborhood
of the origin.
\medskip
\item{10.} If $f(z)$ is analytic in $|z|<1$ and $|f(z)|\le1$ for all $|z|<1$, show
that $(1-|z|^2)|f'(z)|\le1$ for all $|z|<1$.
\end