Take the 2-minute tour ×
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.

$f$ and $g$ are holomoprphic functions in $G \subset \mathbb C$ and continuous on the boundary of $G$. Prove that $|f| + |g|$ gets its maximum in the boundary of $G$.

I know this has something to do with the maximum principle, but I'd be happy for a hint.

share|improve this question
add comment

2 Answers

up vote 1 down vote accepted

I think $G$ is supposed to be bounded and hence $\overline{G}$ is compact. By continuity of $|f|+|g|$ and compactness of $\overline{G}$, there exists $z_0\in\overline{G}$ and $M\ge 0$, such that $|f|+|g|$ attains its maximum $M$ at $z_0$. Let $a,b\in\mathbb{C}$ with $|a|=|b|=1$, such that $|f(z_0)|=af(z_0)$ and $|g(z_0)|=bg(z_0)$. Define $h=af+bg$. Then $h$ is holomorhic on $G$ and continous on $\overline{G}$. Moreover, $|h(z)|\le M$ on $\overline{G}$ and $h(z_0)=M$. Then the conclusion follows from maximum modulus principle.

share|improve this answer
add comment

If $\omega \in \mathbb{C}$ and $|\omega| = 1$ then $f + \omega g$ is holomorphic on $G$ and $|f + \omega g| \leq |f| + |g|$. Given some fixed $z \in G$ there is such an omega (depending on $z$) such that $|f(z) + \omega g(z)| = |f(z)| + |g(z)|$. Now apply the maximum modulus principle to $f + \omega g$.

share|improve this answer
add comment

Your Answer

 
discard

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.