Lindelöf's maximum principle

Remarks

Proof idea: let $d$ be the diameter of $\Omega$ and let $\varepsilon > 0$. The function $v = u + \varepsilon \sum_{p \in F} \log(\abs{z - p}/d)$ is subharmonic, and $v \le u$ on $\Omega$ because each logarithm is negative there. At each $p \in F$, $v \to -\infty$ since $u$ is bounded above; so $\limsup v \le 0$ at every boundary point. The maximum principle for subharmonic functions gives $v \le 0$, and letting $\varepsilon \to 0$ gives $u \le 0$. The bound from above is needed. On the punctured disc $\Omega = \set{0 < \abs{z} < 1}$ with $F = \set{0}$, the function $u(z) = -\log\abs{z}$ is harmonic, tends to $0$ on the unit circle, and is positive on $\Omega$. The finite set $F$ may be replaced by any polar set in $\partial\Omega$, with the same conclusion.

Going deeper

For a student

The lemma is the bounded form of the Phragmén–Lindelöf principle: a growth condition lets one ignore a small part of the boundary. Applied to $\log\abs{f}$ for a bounded holomorphic $f$, it shows that $\abs{f} \le 1$ on $\Omega$ when $\limsup \abs{f} \le 1$ at all but finitely many boundary points. A related result for holomorphic functions on a half-strip is Lindelöf's theorem. In the study of harmonic measure the lemma shows that finite sets of boundary points carry no mass.