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The discussion of the Phragmen-Lindelöf Theorem on page 176 of Titchmarsh's book Theory of Functions starts off with the following result:

Let $C$ be a simple closed contour, and let $f(z)$ be regular inside and on $C$, except at one point $P$ of $C$. Let $|f(z)|\leq M$ on $C$, except at $P$. Suppose further that there is a function $\omega(z)$, regular and not zero in $C$, such that $|\omega(z)| \leq 1$ inside $C$, and such that if $\varepsilon$ is any given positive number, we can find a system of curves, arbitrarily near to $P$ and connecting the two sides of $C$ round $P$, on which $|(\omega(z))^{\varepsilon}f(z)|\leq M$. Then $|f(z)| \leq M$ at all points inside $C$.

Titchmarsh's proof is as follows:

Consider $F(z) = \omega(z)^{\varepsilon}f(z)$. Then $F(z)$ is analytic in $C$. If $z_{0}$ is any point inside $C$, we can, by the hypothesis about $\omega(z)$, find a curve surrounding $z_{0}$ on which $|F(z)| \leq M$. Hence $|F(z_{0})| \leq M$, and so $|f(z_{0})| \leq M|\omega(z_{0})|^{-\varepsilon}$. Making $\varepsilon \rightarrow 0$ yields $|f(z_{0})| \leq M$ which proves the theorem.

My question is: Why can I find a curve surrounding $z_{0}$ for which $|F(z)| \leq M$? I know by how $\omega(z)$ is defined, I can do it for $P$ but what about $z_{0}$? Also is it because the maximum priniciple that $|F(z)| \leq M$ for $z$ in the curve surrounding $z_{0}$ implies that $|F(z_{0})| \leq M$?

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You can obtain the desired curve surrounding $z_0$ from $C$, by replacing an arc of $C$ containing $P$ with a connecting curve (a "bypass") on which $|\omega|^\epsilon |f|\le M$.

Here is a picture, which I hope someone will embed in the post.

And yes, the conclusion $|F(z_0)|\le M$ follows from the maximum principle stated at the beginning of Chapter V.

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