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\section*{Problem Set 4}

\begin{enumerate}[leftmargin=0.7cm]
\item Show $|z|^{\alpha}$ is subharmonic (on $\mathbb{C}$) for every positive real $\alpha$.\vspace{2mm}
\item Let $U=D_1^*$ and $f$ be a continuous function on the boundary (unit circle together with the origin). Then the first 2 steps of Perron's proof go through: that is, if $u$ (function on $D_1^*$) is the pointwise supremum of the functions in $\mathcal{S}$, then $u$ is bounded above and harmonic. Determine $u$ in both of the following cases: 

(a) $f\equiv 1$ on the unit circle and $f(0)=0$; 

(b) $f\equiv 0$ on the unit circle and $f(0)=1$. 

\noindent If $u$ is not in $\mathcal{S}$, exhibit a sequence of functions converging to it (which is guaranteed by the second part of the proof). [Hint: use (1), though you only need that $|z|^{\alpha}$ is subharmonic on $D_1^*$.]\vspace{2mm}
\item Let $U$ be a bounded and simply connected region which has a barrier at each point, and let $\phi$ be a positive (real valued) continuous function on the boundary $\partial U$. Prove that there exists a continuous (complex valued) function $f$ on $\overline{U}$ which is \emph{holomorphic} in $U$ and has $|f|=\phi$ on $\partial U$.\vspace{2mm}
\item Carry out the following outline of proof that a bounded region $U$ with boundary consisting of two $C^1$ Jordan curves (the domain being homeomorphic to an annulus) is biholomorphic to a region of the form $\{z\mid 1<|z|<R\}$.

 (a) Prove that there exists $u\in \mathcal{H}(U)$ such that the limit of $u$ at the outer (resp.~inner) boundary is $1$ (resp.~$0$).

 (b) Choose a fixed loop $\gamma$ going once around the hole counterclockwise. Set $\omega=-u_y dx+u_x dy$. Show $\int_{\gamma}\omega\neq 0$ by supposing otherwise (i.e.~$\int_{\gamma}\omega=0$) and applying the argument principle to $u+\mathbf{i}\int \omega$.

 (c) Pick $\lambda\in \mathbb{R}$ so that $\int_{\gamma}\lambda\omega=2\pi$. Show that $\exp\{\lambda u+\mathbf{i}\int\lambda\omega\}$ is a well-defined, holomorphic, 1-to-1 and onto mapping from $U$ to $\{z \mid 1<|z|<e^{\lambda} \}$.
\end{enumerate}

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