English
If f and another function satisfy the P-L hypotheses in the third quadrant and coincide on the boundary, then they coincide on the closed third quadrant.
Русский
Если функции удовлетворяют условиямв третьем квадранте и совпадают на границе, то совпадают и в закрытом третьем квадранте.
LaTeX
$$$$\\forall z \\in \\{\\Re z \\le 0, \\Im z \\le 0\\}, \\ f(z) = g(z).$$$$
Lean4
/-- **Phragmen-Lindelöf principle** in the third quadrant. Let `f : ℂ → E` be a function such that
* `f` is differentiable in the open third quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * ‖z‖ ^ c)` on the open third quadrant
for some `A`, `B`, and `c < 2`;
* `f` is equal to zero on the boundary of the third quadrant.
Then `f` is equal to zero on the closed third quadrant. -/
theorem eq_zero_on_quadrant_III (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0))
(hB : ∃ c < (2 : ℝ), ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * ‖z‖ ^ c))
(hre : ∀ x : ℝ, x ≤ 0 → f x = 0) (him : ∀ x : ℝ, x ≤ 0 → f (x * I) = 0) : EqOn f 0 {z | z.re ≤ 0 ∧ z.im ≤ 0} :=
fun _z hz =>
norm_le_zero_iff.1 <|
quadrant_III hd hB (fun x hx => norm_le_zero_iff.2 <| hre x hx) (fun x hx => norm_le_zero_iff.2 <| him x hx) hz.1
hz.2
