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

* `f` is differentiable in the open right half-plane and is continuous on its closure;
* `‖f z‖` is bounded from above by `A * exp(B * (abs z) ^ c)` on the open right half-plane
  for some `c < 2`;
* `‖f z‖` is bounded from above by a constant `C` on the imaginary axis;
* `f x → 0` as `x : ℝ` tends to infinity.

Then `‖f z‖` is bounded from above by the same constant on the closed right half-plane.
See also `PhragmenLindelof.right_half_plane_of_bounded_on_real` for a stronger version. -/
theorem right_half_plane_of_tendsto_zero_on_real (hd : DiffContOnCl ℂ f {z | 0 < z.re})
    (hexp : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * abs z ^ c))
    (hre : Tendsto (fun x : ℝ => f x) atTop (𝓝 0)) (him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C)
    (hz : 0 ≤ z.re) : ‖f z‖ ≤ C := by
  /- We are going to apply the Phragmen-Lindelöf principle in the first and fourth quadrants.
    The lemmas immediately imply that for any upper estimate `C'` on `‖f x‖`, `x : ℝ`, `0 ≤ x`,
    the number `max C C'` is an upper estimate on `f` in the whole right half-plane. -/
  revert z
  have hle : ∀ C', (∀ x : ℝ, 0 ≤ x → ‖f x‖ ≤ C') →
      ∀ z : ℂ, 0 ≤ z.re → ‖f z‖ ≤ max C C' := fun C' hC' z hz ↦ by
    rcases hexp with ⟨c, hc, B, hO⟩
    rcases le_total z.im 0 with h | h
    · refine quadrant_IV (hd.mono fun _ => And.left) ⟨c, hc, B, ?_⟩
          (fun x hx => (hC' x hx).trans <| le_max_right _ _)
          (fun x _ => (him x).trans (le_max_left _ _)) hz h
      exact hO.mono (inf_le_inf_left _ <| principal_mono.2 fun _ => And.left)
    · refine' quadrant_I (hd.mono fun _ => And.left) ⟨c, hc, B, ?_⟩
          (fun x hx => (hC' x hx).trans <| le_max_right _ _)
          (fun x _ => (him x).trans (le_max_left _ _)) hz h
      exact hO.mono (inf_le_inf_left _ <| principal_mono.2 fun _ => And.left)
  -- Since `f` is continuous on `Ici 0` and `‖f x‖` tends to zero as `x → ∞`,
  -- the norm `‖f x‖` takes its maximum value at some `x₀ : ℝ`.
  obtain ⟨x₀, hx₀, hmax⟩ : ∃ x : ℝ, 0 ≤ x ∧ ∀ y : ℝ, 0 ≤ y → ‖f y‖ ≤ ‖f x‖ := by
    have hfc : ContinuousOn (fun x : ℝ => f x) (Ici 0) := by
      refine' hd.continuousOn.comp continuous_ofReal.continuousOn fun x hx => _
      rwa [closure_setOf_lt_re]
    by_cases h₀ : ∀ x : ℝ, 0 ≤ x → f x = 0
    · refine' ⟨0, le_rfl, fun y hy => _⟩; rw [h₀ y hy, h₀ 0 le_rfl]
    push_neg at h₀
    rcases h₀ with ⟨x₀, hx₀, hne⟩
    have hlt : ‖(0 : E)‖ < ‖f x₀‖ := by rwa [norm_zero, norm_pos_iff]
    suffices ∀ᶠ x : ℝ in cocompact ℝ ⊓ 𝓟 (Ici 0), ‖f x‖ ≤ ‖f x₀‖ by
      simpa only [exists_prop] using hfc.norm.exists_forall_ge' isClosed_Ici hx₀ this
    rw [Real.cocompact_eq, inf_sup_right, (disjoint_atBot_principal_Ici (0 : ℝ)).eq_bot,
      bot_sup_eq]
    exact (hre.norm.eventually <| ge_mem_nhds hlt).filter_mono inf_le_left
  rcases le_or_lt ‖f x₀‖ C with h | h
  ·-- If `‖f x₀‖ ≤ C`, then `hle` implies the required estimate
    

Proof state:
case intro.intro.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hre : Tendsto (fun x => f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
h : ‖f ↑x₀‖ ≤ C
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C

Next tactic:
simpa only [max_eq_left h] using hle _ hmax