English
If f and g obey the growth bounds in quadrant IV and agree on the boundary, they agree on the closed quadrant.
Русский
Если f и g удовлетворяют ростовому условию во IV квадранте и совпадают на границе, они совпадают на всей закрытой четвертой четверти.
LaTeX
$$$$\\forall z \\in \\{ \\Re z \\ge 0, \\Im z \\le 0 \\}, \\ f(z) = g(z).$$$$
Lean4
/-- **Phragmen-Lindelöf principle** in the fourth quadrant. Let `f g : ℂ → E` be functions such that
* `f` and `g` are differentiable in the open fourth quadrant and are continuous on its closure;
* `‖f z‖` and `‖g z‖` are bounded from above by `A * exp(B * ‖z‖ ^ c)` on the open fourth
quadrant for some `A`, `B`, and `c < 2`;
* `f` is equal to `g` on the boundary of the fourth quadrant.
Then `f` is equal to `g` on the closed fourth quadrant. -/
theorem eqOn_quadrant_IV (hdf : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0))
(hBf : ∃ c < (2 : ℝ), ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z => expR (B * ‖z‖ ^ c))
(hdg : DiffContOnCl ℂ g (Ioi 0 ×ℂ Iio 0))
(hBg : ∃ c < (2 : ℝ), ∃ B, g =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z => expR (B * ‖z‖ ^ c))
(hre : ∀ x : ℝ, 0 ≤ x → f x = g x) (him : ∀ x : ℝ, x ≤ 0 → f (x * I) = g (x * I)) :
EqOn f g {z | 0 ≤ z.re ∧ z.im ≤ 0} := fun _z hz =>
sub_eq_zero.1 <|
eq_zero_on_quadrant_IV (hdf.sub hdg) (isBigO_sub_exp_rpow hBf hBg) (fun x hx => sub_eq_zero.2 <| hre x hx)
(fun x hx => sub_eq_zero.2 <| him x hx) hz
