The open mapping theorem for holomorphic functions #

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This file proves the open mapping theorem for holomorphic functions, namely that an analytic function on a preconnected set of the complex plane is either constant or open. The main step is to show a local version of the theorem that states that if f is analytic at a point z₀, then either it is constant in a neighborhood of z₀ or it maps any neighborhood of z₀ to a neighborhood of its image f z₀. The results extend in higher dimension to g : E → ℂ.

The proof of the local version on ℂ goes through two main steps: first, assuming that the function is not constant around z₀, use the isolated zero principle to show that ‖f z‖ is bounded below on a small sphere z₀ r around z₀, and then use the maximum principle applied to the auxiliary function (λ z, ‖f z - v‖) to show that any v close enough to f z₀ is in f '' ball z₀ r. That second step is implemented in diff_cont_on_cl.ball_subset_image_closed_ball.

Main results #

analytic_at.eventually_constant_or_nhds_le_map_nhds is the local version of the open mapping theorem around a point;
analytic_on.is_constant_or_is_open is the open mapping theorem on a connected open set.

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theorem diff_cont_on_cl.ball_subset_image_closed_ball {f : ℂ → ℂ} {z₀ : ℂ} {ε r : ℝ} (h : diff_cont_on_cl ℂ f (metric.ball z₀ r)) (hr : 0 < r) (hf : ∀ (z : ℂ), z ∈ metric.sphere z₀ r → ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ (z : ℂ) in nhds z₀, f z ≠ f z₀) :

metric.ball (f z₀) (ε / 2) ⊆ f '' metric.closed_ball z₀ r

If the modulus of a holomorphic function f is bounded below by ε on a circle, then its range contains a disk of radius ε / 2.

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theorem analytic_at.eventually_constant_or_nhds_le_map_nhds_aux {f : ℂ → ℂ} {z₀ : ℂ} (hf : analytic_at ℂ f z₀) :

(∀ᶠ (z : ℂ) in nhds z₀, f z = f z₀) ∨ nhds (f z₀) ≤ filter.map f (nhds z₀)

A function f : ℂ → ℂ which is analytic at a point z₀ is either constant in a neighborhood of z₀, or behaves locally like an open function (in the sense that the image of every neighborhood of z₀ is a neighborhood of f z₀, as in is_open_map_iff_nhds_le). For a function f : E → ℂ the same result holds, see analytic_at.eventually_constant_or_nhds_le_map_nhds.

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theorem analytic_at.eventually_constant_or_nhds_le_map_nhds {E : Type u_1} [normed_add_comm_group E] [normed_space ℂ E] {g : E → ℂ} {z₀ : E} (hg : analytic_at ℂ g z₀) :

(∀ᶠ (z : E) in nhds z₀, g z = g z₀) ∨ nhds (g z₀) ≤ filter.map g (nhds z₀)

The open mapping theorem for holomorphic functions, local version: is a function g : E → ℂ is analytic at a point z₀, then either it is constant in a neighborhood of z₀, or it maps every neighborhood of z₀ to a neighborhood of z₀. For the particular case of a holomorphic function on ℂ, see analytic_at.eventually_constant_or_nhds_le_map_nhds_aux.

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theorem analytic_on.is_constant_or_is_open {E : Type u_1} [normed_add_comm_group E] [normed_space ℂ E] {U : set E} {g : E → ℂ} (hg : analytic_on ℂ g U) (hU : is_preconnected U) :

(∃ (w : ℂ), ∀ (z : E), z ∈ U → g z = w) ∨ ∀ (s : set E), s ⊆ U → is_open s → is_open (g '' s)

The open mapping theorem for holomorphic functions, global version: if a function g : E → ℂ is analytic on a connected set U, then either it is constant on U, or it is open on U (in the sense that it maps any open set contained in U to an open set in ℂ).