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Boundary maximum modulus principle on a bounded domain

Statement

If $Ω$ is a bounded complex domain and $f$ is continuous on $\overline{Ω}$ and holomorphic on $Ω$ , then $∣ f ∣$ attains its maximum on $\partial Ω$ .

Equivalently, there is $ζ \in \partial Ω$ such that $∣ f (z) ∣ \leq ∣ f (ζ) ∣ (z \in \overline{Ω}) .$

Facts & Assumptions

Given: A bounded complex domain $Ω$ and a continuous function $f : \overline{Ω} \to C$ whose restriction to $Ω$ is holomorphic. The complex-plane topology and Euclidean-plane topology agree ( $C = R [x] / (x^{2} + 1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves).

[L1]

If the modulus of a holomorphic function on a complex domain has an interior local maximum, then the function is constant (Local maximum modulus principle).

[L2]

A subset of $R^{n}$ is compact exactly when it is closed and bounded, for $n \geq 1$ (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[L3]

A continuous real-valued function on a nonempty compact metric space has a maximum and a minimum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[L4]

The Euclidean plane $R^{2}$ is connected ( $R^{n}$ is polygonally connected, connected, locally path-connected and locally connected).

Proof

technique · direct

1.1L2L3givenalgebra

The closure $\overline{Ω}$ is nonempty, closed, and bounded in the Euclidean plane, hence compact by [L2]. The reverse triangle inequality and continuity of $f$ make $∣ f ∣$ continuous there, so [L3] gives a maximizer $z_{0} \in \overline{Ω}$ .

2.1step 1.1L1

If $z_{0} \in Ω$ , then $∣ f ∣$ has an interior local maximum, and [L1] makes $f$ constant on $Ω$ .

3.1step 1.1step 2.1L4given

If $z_{0} \notin Ω$ , then $z_{0} \in \partial Ω$ . In the remaining branch $f$ is constant on $Ω$ by step 2.1 and hence on $\overline{Ω}$ by continuity. The boundary is nonempty: otherwise the nonempty open set $Ω$ would also be closed in the connected plane [L4] and therefore equal the unbounded plane. Thus any boundary point has the same modulus as $z_{0}$ .

4.1step 3.1∎

In either branch, a point of $\partial Ω$ carries the global maximum of $∣ f ∣$ on $\overline{Ω}$ .

Depends on

Used by

Dependency tree · two levels

54 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, Corollary 3.3.7 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, Theorem 1.15 (standard reference, not scraped)
J. A. Tropp, Matrix Analysis, Theorem 7.12 (standard reference, not scraped)