\documentclass[12pt,oneside,english]{amsart}
\renewcommand{\familydefault}{\rmdefault}
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}
\usepackage{amsmath,amsfonts,amstext,amssymb,mathtools}
\usepackage{mathrsfs}
\usepackage{enumitem}
\usepackage{bm}
\usepackage[all]{xy}
\usepackage{fullpage}

\def\RR{\mathbb{R}}
\def\CC{\mathbb{C}}
\def\ZZ{\mathbb{Z}}
\def\vf{\varphi}

\makeatletter

\begin{document}

\section*{Problem Set 10}

\begin{enumerate}
\item Find polynomials (for each $n$) realizing the upper bound (namely, $\tfrac{n}{n+1}$) on $\sigma(P)$ in the Smale Conjecture.  (See the end of Lecture 30.)
\item Examine the proof of Bloch's theorem (Lecture 30) to prove that $L\geq \tfrac{1}{24}$.
\item Prove that the power-series coefficients $a_n$, viewed as functions on the set $\mathcal{S}$ of schlicht functions, are continuous functions in the normal topology. (This implies that there must exist universal upper bounds on them, as $\mathcal{S}$ is compact in this topology.)
\item Let $\Omega\subsetneq \CC$ be a simply connected domain, $\alpha\in \Omega$ a point.  The Riemann Mapping Theorem guarantees a conformal isomorphism $F\colon \Omega\to D_1$ with $F(\alpha)=0$. Show that $\tfrac{1}{4\cdot d(\alpha,\partial\Omega)}\leq |F'(\alpha)|\leq \tfrac{1}{d(\alpha,\partial\Omega)}$. (You'll need to use K\"obe $\tfrac{1}{4}$, which will appear in Monday's notes.  Also Schwarz lemma.)
\end{enumerate}

\end{document}