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\begin{center}{{\bf Complex analysis preliminary examination \\ August 31, 2000}} \end{center}

\vspace{3cm}

1. Let $f$ be a harmonic function in the unit disk. Given that $e^f$ is harmonic prove that eithr $f$ is holomorphic, or
$\bar{f}$ is holomorphic.

\vspace{1cm}

2. Let $f_n$, $n=0,1,...$ be a sequence of functions analytic in a closed bounded domain $\bar{\Omega } $ with smooth boundary.
Suppose that $f_n\to f_0$ as $n\to \infty $ uniformly on compact subsets of $\Omega $. Does this imply that
$$
\int_{\partial \Omega }|f_n(z)|ds(z) \to \int_{\partial\Omega }|f_0(z)|ds(z),
$$
where $ds$ is the linear Lebesgue measure on $\partial \Omega $. Prove, or give a counterexample.

\vspace{1cm}

3. If $f$ is analytic in the unit disk $\Delta $, continuous in $\bar{\Delta }$ and maps $\Delta $ into itself, prove that for every point $a\in \Delta $
$$
f^{(3)}(a)\leq \frac{6(1+|a|^2)}{(1-|a|^2)^3}.
$$
Hint: Use Cauchy formula. 

\vspace{1cm}

4. Prove that
$$
\int_{0}^{\infty }\frac{(\log x)^2}{1+x^2}dx=\frac{\pi ^3}{8}.
$$

\vspace{1cm}

5. Let $B$ be a finite Blaschke product. Prove that
$$
\frac{1}{2\pi i}\int_{|z|=1}\frac{dz}{B(z)}=\bar{B^{\prime }(0)}.
$$

\vspace{1cm}

6. Let $f$ be analytic and absolutely integrable with respect to the Lebesgue area measure $dA=dxdy$ in the unit disk $\Delta $.
Prove that for $a\in \Delta $
$$
\int_{\Delta }f(z)\frac{\bar{z}}{(1-a\bar{z})^3}dA(z)=\frac{1}{2}f^{\prime }(a).
$$

\vspace{1cm}

7. Let $f$ be analytic in the closed unt disk and $|f(z)|$ is  constant for \newline $|z|=1$. Prove that $arg(f(e^{i\theta }))$ is
a monotone function of $\theta $.

\vspace{1cm}

8. Let $f$ be a smooth bounded function in the unit disk $\Delta $ and  $F(z)$ be given by
$$
F(z)=\frac{1}{\pi }\int_{\Delta }\frac{f(w)dA(w)}{w-z}.
$$
Prove that F(z) is analytic outside of $\bar{\Delta }$ and for every $z\in \Delta $
$$
\frac{\partial F}{\partial \bar{z}}=f(z).
$$
Hint: Use Cauchy-Green formula.

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