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\noindent{\Large{\textsc{Complex Variables\hfill Assignment 8.2  \\ Spring 2020 \hfill Due April 1}}}

\large

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\begin{exercise}
Textbook exercise III.5.2.
\end{exercise}

\begin{exercise}
Let $f(z)$ be analytic and bounded on the right half-plane $\operatorname{Re} z > 0$.  Suppose $f(z)$ extends continuously to the imaginary axis and satisfies $|f(iy)| \le M$ for all $y \in \R$.  We will show that $|f(z)| \le M$ throughout $\operatorname{Re} z > 0$.  

Fix $\epsilon > 0$ and let $\displaystyle g_{\epsilon}(z) = \frac{f(z)}{(1+z)^{\epsilon}}$, where we are using the principal branch of $w^{\epsilon}$.  For $R>1$ let $\Omega_R$ denote $\{\operatorname{Re} z > 0\} \cap \{|z + 1| <R \}$ and let $C_R = \partial \Omega_R$.  

\begin{itemize}

\item[\bf a.]  Show that if $R$ is sufficiently large, then $|g_{\epsilon}(z)| \le M$ on $C_R$.  Conclude that $|g_{\epsilon}(z)| \le M$ on $\Omega_R$, when $R$ is large enough.

\item[\bf b.]  Use part {\bf a} to prove that $|g_{\epsilon}(z)| \le M$ everywhere on the right half-plane $\operatorname{Re} z > 0$.

\item[\bf c.]  Use part {\bf b} to show that $|f(z)| \le M$ throughout the half-plane, by letting $\epsilon \to 0^+$ pointwise.

\end{itemize}

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