English
Under the same hypotheses, f and g coincide on the horizontal strip closure.
Русский
При тех же условиях f и g совпадают на замкнутой горизонтальной полосе.
LaTeX
$$EqOn f g (Set.preimage Complex.im (Set.Icc a b))$$
Lean4
/-- **Phragmen-Lindelöf principle** in a strip `U = {z : ℂ | a < re z < b}`.
Let `f : ℂ → E` be a function such that
* `f` is differentiable on `U` and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * exp(c * |im z|))` on `U` for some `c < π / (b - a)`;
* `f z = 0` on the boundary of `U`.
Then `f` is equal to zero on the closed strip `{z : ℂ | a ≤ re z ≤ b}`.
-/
theorem eq_zero_on_vertical_strip (hd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b))
(hB :
∃ c < π / (b - a),
∃ B, f =O[comap (_root_.abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.im|)))
(ha : ∀ z : ℂ, re z = a → f z = 0) (hb : ∀ z : ℂ, re z = b → f z = 0) : EqOn f 0 (re ⁻¹' Icc a b) := fun _z hz =>
norm_le_zero_iff.1 <|
vertical_strip hd hB (fun z hz => (ha z hz).symm ▸ norm_zero.le) (fun z hz => (hb z hz).symm ▸ norm_zero.le) hz.1
hz.2
