/-- **Phragmen-Lindelöf principle** in a strip `U = {z : ℂ | a < im 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 * |re z|))` on `U` for some `c < π / (b - a)`;
* `‖f z‖` is bounded from above by a constant `C` on the boundary of `U`.

Then `‖f z‖` is bounded by the same constant on the closed strip
`{z : ℂ | a ≤ im z ≤ b}`. Moreover, it suffices to verify the second assumption
only for sufficiently large values of `|re z|`.
-/
theorem horizontal_strip (hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
    (hB : ∃ c < π / (b - a), ∃ B,
      f =O[comap (Abs.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|)))
    (hle_a : ∀ z : ℂ, im z = a → ‖f z‖ ≤ C) (hle_b : ∀ z, im z = b → ‖f z‖ ≤ C) (hza : a ≤ im z)
    (hzb : im z ≤ b) : ‖f z‖ ≤ C := by
  -- If `im z = a` or `im z = b`, then we apply `hle_a` or `hle_b`, otherwise `im z ∈ Ioo a b`.
  rw [le_iff_eq_or_lt] at hza hzb
  cases' hza with hza hza; · exact hle_a _ hza.symm
  cases' hzb with hzb hzb; · exact hle_b _ hzb
  wlog hC₀ : 0 < C generalizing C
  · refine' le_of_forall_le_of_dense fun C' hC' => this (fun w hw => _) (fun w hw => _) _
    · exact (hle_a _ hw).trans hC'.le
    · exact (hle_b _ hw).trans hC'.le
    · refine' ((norm_nonneg (f (a * I))).trans (hle_a _ _)).trans_lt hC'
      rw [mul_I_im, ofReal_re]
  -- After a change of variables, we deal with the strip `a - b < im z < a + b` instead
  -- of `a < im z < b`
  obtain ⟨a, b, rfl, rfl⟩ : ∃ a' b', a = a' - b' ∧ b = a' + b' :=
    ⟨(a + b) / 2, (b - a) / 2, by ring, by ring⟩
  have hab : a - b < a + b := hza.trans hzb
  have hb : 0 < b := by simpa only [sub_eq_add_neg, add_lt_add_iff_left, neg_lt_self_iff] using hab
  rw [add_sub_sub_cancel, ← two_mul, div_mul_eq_div_div] at hB
  have hπb : 0 < π / 2 / b := div_pos Real.pi_div_two_pos hb
  -- Choose some `c B : ℝ` satisfying `hB`, then choose `max c 0 < d < π / 2 / b`.
  rcases hB with ⟨c, hc, B, hO⟩
  obtain ⟨d, ⟨hcd, hd₀⟩, hd⟩ : ∃ d, (c < d ∧ 0 < d) ∧ d < π / 2 / b := by
    simpa only [max_lt_iff] using exists_between (max_lt hc hπb)
  have hb' : d * b < π / 2 := (lt_div_iff hb).1 hd
  set aff := (fun w => d * (w - a * I) : ℂ → ℂ)
  set g := fun (ε : ℝ) (w : ℂ) => exp (ε * (exp (aff w) + exp (-aff w)))
  /- Since `g ε z → 1` as `ε → 0⁻`, it suffices to prove that `‖g ε z • f z‖ ≤ C`
    for all negative `ε`. -/
  suffices : ∀ᶠ ε : ℝ in 𝓝[<] (0 : ℝ), ‖g ε z • f z‖ ≤ C
  · refine' le_of_tendsto (Tendsto.mono_left _ nhdsWithin_le_nhds) this
    apply ((continuous_ofReal.mul continuous_const).cexp.smul continuous_const).norm.tendsto'
    simp
  filter_upwards [self_mem_nhdsWithin] with ε ε₀; change ε < 0 at ε₀
  -- An upper estimate on `‖g ε w‖` that will be used in two branches of the proof.
  obtain ⟨δ, δ₀, hδ⟩ :
    ∃ δ : ℝ,
      δ < 0 ∧ ∀ ⦃w⦄, im w ∈ Icc (a - b) (a + b) → abs (g ε w) ≤ expR (δ * expR (d * |re w|)) := by
    refine'
      ⟨ε * Real.cos (d * b),
        mul_neg_of_neg_of_pos ε₀
          (Real.cos_pos_of_mem_Ioo <| abs_lt.1 <| (abs_of_pos (mul_pos hd₀ hb)).symm ▸ hb'),
        fun w hw => _⟩
    replace hw : |im (aff w)| ≤ d * b
    · rw [← Real.closedBall_eq_Icc] at hw
      rwa [ofReal_mul_im, sub_im, mul_I_im, ofReal_re, _root_.abs_mul, abs_of_pos hd₀,
        mul_le_mul_left hd₀]
    simpa only [ofReal_mul_re, _root_.abs_mul, abs_of_pos hd₀, sub_re, mul_I_re, ofReal_im,
      zero_mul, neg_zero, sub_zero] using
      abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le ε₀.le hw hb'.le
  -- `abs (g ε w) ≤ 1` on the lines `w.im = a ± b` (actually, it holds everywhere in the strip)
  have hg₁ : ∀ w, im w = a - b ∨ im w = a + b → abs (g ε w) ≤ 1 := by
    refine' fun w hw => (hδ <| hw.by_cases _ _).trans (Real.exp_le_one_iff.2 _)
    exacts [fun h => h.symm ▸ left_mem_Icc.2 hab.le, fun h => h.symm ▸ right_mem_Icc.2 hab.le,
      mul_nonpos_of_nonpos_of_nonneg δ₀.le (Real.exp_pos _).le]
  /- Our apriori estimate on `f` implies that `g ε w • f w → 0` as `|w.re| → ∞` along the strip. In
    particular, its norm is less than or equal to `C` for sufficiently large `|w.re|`. -/
  obtain ⟨R, hzR, hR⟩ :
    ∃ R : ℝ, |z.re| < R ∧ ∀ w, |re w| = R → im w ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C := by
    refine' ((eventually_gt_atTop _).and _).exists
    rcases hO.exists_pos with ⟨A, hA₀, hA⟩
    

Proof state:
case intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f g✝ : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (Abs.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))
d : ℝ
hd : d < π / 2 / b
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w => ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w => cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → Complex.abs (g ε w) ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → Complex.abs (g ε w) ≤ 1
A : ℝ
hA₀ : A > 0
hA : IsBigOWith A (comap (Abs.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))) f fun z => expR (B * expR (c * |z.re|))
⊢ ∀ᶠ (R : ℝ) in atTop, ∀ (w : ℂ), |w.re| = R → w.im ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C

Next tactic:
simp only [isBigOWith_iff, eventually_inf_principal, eventually_comap, mem_Ioo, ← abs_lt,
      mem_preimage, (· ∘ ·), Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] at hA