\documentclass[12pt,oneside,english]{amsart}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage{amstext}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{fullpage}

\begin{document}

\section*{Problem set 11}
Exercises (3)-(4) prove the theorem of Nevanlinna mentioned at the end of Lecture 33.
\begin{enumerate}
\item Prove that the universal cover of $\mathbb{C}^{*}\backslash[1,\infty)$
cannot be $\mathbb{C}$ or $\mathbb{P}^{1}$.
\item If $\Gamma$ is the fundamental group (the group of homotopy classes
of loops) of a Riemann surface $M$ having $\mathfrak{H}$ as universal
covering surface, then $M\cong\mathfrak{H}/\Gamma$. Prove that 2
Riemann surfaces $M_{i}\cong\mathfrak{H}/\Gamma_{i}$ ($i=1,2$) are
isomorphic if and only if $\Gamma_{1}$ and $\Gamma_{2}$ are conjugate
in $\mathrm{PGL}(2,\mathbb{R})$.
\item Let $\mathcal{S}^{*}:=\{f\in\mathcal{S}\,|\,f(D)\text{ is starlike}\}$.
(A set $E$ is starlike if for every $z\in E$, $tz\in E$ for every
$t\in[0,1]$.) Prove that $Re(z\frac{f'}{f})\geq0$ for an $f\in\mathcal{S}^{*}$
by following the steps:\\
(a) Show $\frac{\partial}{\partial\theta}\arg(f(re^{i\theta}))>0$,
for any $r\in(0,1)$. {[}Hint: first show that $f(\overline{D}_{r})$
is starlike, making use of the function $f^{-1}(rf(z))$ and Schwarz's
Lemma.{]}\\
(b) Show that $\frac{\partial}{\partial\theta}\arg f=Re(z\frac{f'}{f})$,
for $|z|=r\in(0,1)$.
\item Again let $f\in\mathcal{S}^{*}$, and define $g(z):=\int_{0}^{z}f(w)\frac{dw}{w}\in \mathrm{Hol}(D)$.\\
(a) Show that $Re(1+z\frac{g''}{g'})\geq0$. {[}Hint: use problem
3.{]}\\
(b) Prove $g(D)$ is convex. {[}Hint: show $\frac{\partial}{\partial\theta}\arg(\frac{\partial}{\partial\theta}g(re^{i\theta}))\geq0$,
and interpret this geometrically.{]}\\
(c) Writing $g(z)=z+\sum_{n\geq2}b_{n}z^{n}$, show that each $|b_{m}|\geq1$.
{[}Hint: define $G(z):=\frac{1}{m}\sum_{j=1}^{m}g(\zeta_{m}^{j}z)$,
where $\zeta_{m}=e^{2\pi i/m}$. Explain why $G(D)\subset g(D)$,
then consider $G(z):=g^{-1}(G(z))$ and compare the lowest order terms
in the expansions of $g(h(z))$ and $G(z)$.{]}\\
(d) Writing $f(z):=z+\sum_{n\geq2}a_{n}z^{n}$, deduce that $|a_{m}|\leq m$
for each $m$.
\end{enumerate}

\end{document}
