Maximum modulus principle #

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In this file we prove several versions of the maximum modulus principle. There are several statements that can be called "the maximum modulus principle" for maps between normed complex spaces. They differ by assumptions on the domain (any space, a nontrivial space, a finite dimensional space), assumptions on the codomain (any space, a strictly convex space), and by conclusion (either equality of norms or of the values of the function).

Main results #

Theorems for any codomain #

Consider a function f : E → F that is complex differentiable on a set s, is continuous on its closure, and ‖f x‖ has a maximum on s at c. We prove the following theorems.

• complex.norm_eq_on_closed_ball_of_is_max_on: if s = metric.ball c r, then ‖f x‖ = ‖f c‖ for any x from the corresponding closed ball;

• complex.norm_eq_norm_of_is_max_on_of_ball_subset: if metric.ball c (dist w c) ⊆ s, then ‖f w‖ = ‖f c‖;

• complex.norm_eq_on_of_is_preconnected_of_is_max_on: if U is an open (pre)connected set, f is complex differentiable on U, and ‖f x‖ has a maximum on U at c ∈ U, then ‖f x‖ = ‖f c‖ for all x ∈ U;

• complex.norm_eq_on_closure_of_is_preconnected_of_is_max_on: if s is open and (pre)connected and c ∈ s, then ‖f x‖ = ‖f c‖ for all x ∈ closure s;

• complex.norm_eventually_eq_of_is_local_max: if f is complex differentiable in a neighborhood of c and ‖f x‖ has a local maximum at c, then ‖f x‖ is locally a constant in a neighborhood of c.

Theorems for a strictly convex codomain #

If the codomain F is a strictly convex space, then in the lemmas from the previous section we can prove f w = f c instead of ‖f w‖ = ‖f c‖, see complex.eq_on_of_is_preconnected_of_is_max_on_norm, complex.eq_on_closure_of_is_preconnected_of_is_max_on_norm, complex.eq_of_is_max_on_of_ball_subset, complex.eq_on_closed_ball_of_is_max_on_norm, and complex.eventually_eq_of_is_local_max_norm.

Values on the frontier #

Finally, we prove some corollaries that relate the (norm of the) values of a function on a set to its values on the frontier of the set. All these lemmas assume that E is a nontrivial space. In this section f g : E → F are functions that are complex differentiable on a bounded set s and are continuous on its closure. We prove the following theorems.

• complex.exists_mem_frontier_is_max_on_norm: If E is a finite dimensional space and s is a nonempty bounded set, then there exists a point z ∈ frontier s such that λ z, ‖f z‖ takes it maximum value on closure s at z.

• complex.norm_le_of_forall_mem_frontier_norm_le: if ‖f z‖ ≤ C for all z ∈ frontier s, then ‖f z‖ ≤ C for all z ∈ s; note that this theorem does not require E to be a finite dimensional space.

• complex.eq_on_closure_of_eq_on_frontier: if f x = g x on the frontier of s, then f x = g x on closure s;

• complex.eq_on_of_eq_on_frontier: if f x = g x on the frontier of s, then f x = g x on s.

Tags #

maximum modulus principle, complex analysis

Auxiliary lemmas #

We split the proof into a series of lemmas. First we prove the principle for a function f : ℂ → F with an additional assumption that F is a complete space, then drop unneeded assumptions one by one.

The lemmas with names *_auxₙ are considered to be private and should not be used outside of this file.

Now we drop the assumption complete_space F by embedding F into its completion.

Then we replace the assumption is_max_on (norm ∘ f) (closed_ball z r) z with a seemingly weaker assumption is_max_on (norm ∘ f) (ball z r) z.

Maximum modulus principle for any codomain #

If we do not assume that the codomain is a strictly convex space, then we can only claim that the norm ‖f x‖ is locally constant.

Finally, we generalize the theorem from a disk in ℂ to a closed ball in any normed space.

The case of a strictly convex codomain #

If the codomain F is a strictly convex space, then we can claim equalities like f w = f z instead of ‖f w‖ = ‖f z‖.

Instead of repeating the proof starting with lemmas about integrals, we apply a corresponding lemma above twice: for f and for λ x, f x + f c. Then we have ‖f w‖ = ‖f z‖ and ‖f w + f z‖ = ‖f z + f z‖, thus ‖f w + f z‖ = ‖f w‖ + ‖f z‖. This is only possible if f w = f z, see eq_of_norm_eq_of_norm_add_eq.

Maximum on a set vs maximum on its frontier #

In this section we prove corollaries of the maximum modulus principle that relate the values of a function on a set to its values on the frontier of this set.