Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $|F(z_{0})| \leq M$?

share|cite|improve this question
up vote 1 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.