/-- **Phragmen-Lindelöf principle** in the first quadrant. Let `f : ℂ → E` be a function such that

* `f` is differentiable in the open first quadrant and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open first quadrant
  for some `c < 2`;
* `‖f z‖` is bounded from above by a constant `C` on the boundary of the first quadrant.

Then `‖f z‖` is bounded from above by the same constant on the closed first quadrant. -/
nonrec theorem quadrant_I (hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, 0 ≤ x → ‖f x‖ ≤ C) (him : ∀ x : ℝ, 0 ≤ x → ‖f (x * I)‖ ≤ C) (hz_re : 0 ≤ z.re)
    (hz_im : 0 ≤ z.im) : ‖f z‖ ≤ C := by
  -- The case `z = 0` is trivial.
  rcases eq_or_ne z 0 with (rfl | hzne);
  · exact hre 0 le_rfl
  -- Otherwise, `z = e ^ ζ` for some `ζ : ℂ`, `0 < Im ζ < π / 2`.
  obtain ⟨ζ, hζ, rfl⟩ : ∃ ζ : ℂ, ζ.im ∈ Icc 0 (π / 2) ∧ exp ζ = z := by
    refine' ⟨log z, _, exp_log hzne⟩
    rw [log_im]
    exact ⟨arg_nonneg_iff.2 hz_im, arg_le_pi_div_two_iff.2 (Or.inl hz_re)⟩
  -- porting note: failed to clear `clear hz_re hz_im hzne`
  -- We are going to apply `PhragmenLindelof.horizontal_strip` to `f ∘ Complex.exp` and `ζ`.
  change ‖(f ∘ exp) ζ‖ ≤ C
  have H : MapsTo exp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0) := fun z hz ↦ by
    rw [mem_reProdIm, exp_re, exp_im, mem_Ioi, mem_Ioi]
    have : 0 < Real.cos z.im := Real.cos_pos_of_mem_Ioo ⟨by linarith [hz.1, hz.2], hz.2⟩
    have : 0 < Real.sin z.im :=
      Real.sin_pos_of_mem_Ioo ⟨hz.1, hz.2.trans (half_lt_self Real.pi_pos)⟩
    constructor <;> positivity
  refine' horizontal_strip (hd.comp differentiable_exp.diffContOnCl H) _ _ _ hζ.1 hζ.2
  -- porting note: failed to clear hζ ζ
  · -- The estimate `hB` on `f` implies the required estimate on
    -- `f ∘ exp` with the same `c` and `B' = max B 0`.
    rw [sub_zero, div_div_cancel' Real.pi_pos.ne']
    rcases hB with ⟨c, hc, B, hO⟩
    refine' ⟨c, hc, max B 0, _⟩
    rw [← comap_comap, comap_abs_atTop, comap_sup, inf_sup_right]
    -- We prove separately the estimates as `ζ.re → ∞` and as `ζ.re → -∞`
    refine' IsBigO.sup _
      ((hO.comp_tendsto <| tendsto_exp_comap_re_atTop.inf H.tendsto).trans <| .of_bound 1 _)
    · -- For the estimate as `ζ.re → -∞`, note that `f` is continuous within the first quadrant at
      -- zero, hence `f (exp ζ)` has a limit as `ζ.re → -∞`, `0 < ζ.im < π / 2`.
      have hc : ContinuousWithinAt f (Ioi 0 ×ℂ Ioi 0) 0 := by
        refine' (hd.continuousOn _ _).mono subset_closure
        simp [closure_reProdIm, mem_reProdIm]
      refine'
        ((hc.tendsto.comp <| tendsto_exp_comap_re_atBot.inf H.tendsto).isBigO_one ℝ).trans
          (isBigO_of_le _ fun w => _)
      rw [norm_one, Real.norm_of_nonneg (Real.exp_pos _).le, Real.one_le_exp_iff]
      exact mul_nonneg (le_max_right _ _) (Real.exp_pos _).le
    · -- For the estimate as `ζ.re → ∞`, we reuse the upper estimate on `f`
      simp only [eventually_inf_principal, eventually_comap, comp_apply, one_mul,
        Real.norm_of_nonneg (Real.exp_pos _).le, abs_exp, ← Real.exp_mul, Real.exp_le_exp]
      refine' (eventually_ge_atTop 0).mono fun x hx z hz _ => _
      rw [hz, _root_.abs_of_nonneg hx, mul_comm _ c]
      exact mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.exp_pos _).le
  · -- If `ζ.im = 0`, then `Complex.exp ζ` is a positive real number
    intro ζ hζ; lift ζ to ℝ using hζ
    rw [comp_apply, ← ofReal_exp]
    exact hre _ (Real.exp_pos _).le
  · -- If `ζ.im = π / 2`, then `Complex.exp ζ` is a purely imaginary number with positive `im`
    intro ζ hζ
    

Proof state:
case inr.intro.intro.refine'_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * Complex.abs z ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ✝ : ℂ
hζ✝ : ζ✝.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ✝).re
hz_im : 0 ≤ (cexp ζ✝).im
hzne : cexp ζ✝ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
ζ : ℂ
hζ : ζ.im = π / 2
⊢ ‖(f ∘ cexp) ζ‖ ≤ C

Next tactic:
rw [← re_add_im ζ, hζ, comp_apply, exp_add_mul_I, ← ofReal_cos, ← ofReal_sin,
      Real.cos_pi_div_two, Real.sin_pi_div_two, ofReal_zero, ofReal_one, one_mul, zero_add, ←
      ofReal_exp]