/-- Suppose `f` is locally integrable on `(0, ∞)`, is `O(x ^ (-a))` as `x → ∞`, and is
`O(x ^ (-b))` as `x → 0`. Then its Mellin transform is differentiable on the domain `b < re s < a`,
with derivative equal to the Mellin transform of `log • f`. -/
theorem mellin_hasDerivAt_of_isBigO_rpow [CompleteSpace E] [NormedSpace ℂ E] {a b : ℝ}
    {f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f (Ioi 0)) (hf_top : f =O[atTop] (· ^ (-a)))
    (hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
    MellinConvergent (fun t => log t • f t) s ∧
      HasDerivAt (mellin f) (mellin (fun t => log t • f t) s) s := by
  set F : ℂ → ℝ → E := fun (z : ℂ) (t : ℝ) => (t : ℂ) ^ (z - 1) • f t
  set F' : ℂ → ℝ → E := fun (z : ℂ) (t : ℝ) => ((t : ℂ) ^ (z - 1) * log t) • f t
  -- A convenient radius of ball within which we can uniformly bound the derivative.
  obtain ⟨v, hv0, hv1, hv2⟩ : ∃ v : ℝ, 0 < v ∧ v < s.re - b ∧ v < a - s.re := by
    obtain ⟨w, hw1, hw2⟩ := exists_between (sub_pos.mpr hs_top)
    obtain ⟨w', hw1', hw2'⟩ := exists_between (sub_pos.mpr hs_bot)
    exact
      ⟨min w w', lt_min hw1 hw1', (min_le_right _ _).trans_lt hw2', (min_le_left _ _).trans_lt hw2⟩
  let bound : ℝ → ℝ := fun t : ℝ => (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
  have h1 : ∀ᶠ z : ℂ in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict <| Ioi 0) := by
    refine' eventually_of_forall fun z => AEStronglyMeasurable.smul _ hfc.aestronglyMeasurable
    refine' ContinuousOn.aestronglyMeasurable _ measurableSet_Ioi
    refine' ContinuousAt.continuousOn fun t ht => _
    exact continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_gt ht)
  have h2 : IntegrableOn (F s) (Ioi (0 : ℝ))
  · exact mellinConvergent_of_isBigO_rpow hfc hf_top hs_top hf_bot hs_bot
  have h3 : AEStronglyMeasurable (F' s) (volume.restrict <| Ioi 0) := by
    apply LocallyIntegrableOn.aestronglyMeasurable
    refine' hfc.continuousOn_smul isOpen_Ioi ((ContinuousAt.continuousOn fun t ht => _).mul _)
    · exact continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_gt ht)
    · refine' continuous_ofReal.comp_continuousOn _
      exact continuousOn_log.mono (subset_compl_singleton_iff.mpr not_mem_Ioi_self)
  have h4 : ∀ᵐ t : ℝ ∂volume.restrict (Ioi 0),
      ∀ z : ℂ, z ∈ Metric.ball s v → ‖F' z t‖ ≤ bound t := by
    refine' (ae_restrict_mem measurableSet_Ioi).mono fun t ht z hz => _
    simp_rw [norm_smul, norm_mul, Complex.norm_eq_abs (log _), Complex.abs_ofReal,
      mul_assoc]
    gcongr
    rw [Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos ht]
    rcases le_or_lt 1 t with h | h
    · refine' le_add_of_le_of_nonneg (rpow_le_rpow_of_exponent_le h _)
        (rpow_nonneg_of_nonneg (zero_le_one.trans h) _)
      rw [sub_re, one_re, sub_le_sub_iff_right]
      rw [mem_ball_iff_norm, Complex.norm_eq_abs] at hz
      have hz' := (re_le_abs _).trans hz.le
      rwa [sub_re, sub_le_iff_le_add'] at hz'
    · refine'
        le_add_of_nonneg_of_le (rpow_pos_of_pos ht _).le (rpow_le_rpow_of_exponent_ge ht h.le _)
      rw [sub_re, one_re, sub_le_iff_le_add, sub_add_cancel]
      rw [mem_ball_iff_norm', Complex.norm_eq_abs] at hz
      have hz' := (re_le_abs _).trans hz.le
      rwa [sub_re, sub_le_iff_le_add, ← sub_le_iff_le_add'] at hz'
  have h5 : IntegrableOn bound (Ioi 0) := by
    simp_rw [add_mul, mul_assoc]
    suffices ∀ {j : ℝ}, b < j → j < a →
        IntegrableOn (fun t : ℝ => t ^ (j - 1) * (|log t| * ‖f t‖)) (Ioi 0) volume by
      refine' Integrable.add (this _ _) (this _ _)
      all_goals linarith
    · intro j hj hj'
      obtain ⟨w, hw1, hw2⟩ := exists_between hj
      obtain ⟨w', hw1', hw2'⟩ := exists_between hj'
      refine' mellin_convergent_of_isBigO_scalar _ _ hw1' _ hw2
      · simp_rw [mul_comm]
        refine' hfc.norm.mul_continuousOn _ isOpen_Ioi
        refine' Continuous.comp_continuousOn continuous_abs (continuousOn_log.mono _)
        exact subset_compl_singleton_iff.mpr not_mem_Ioi_self
      · refine (isBigO_rpow_top_log_smul hw2' hf_top).norm_left.congr_left fun t ↦ ?_
        

Proof state:
case intro.intro.intro.intro.refine'_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0)
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t => ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t => (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t => (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (Measure.restrict volume (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0)
h3 : AEStronglyMeasurable (F' s) (Measure.restrict volume (Ioi 0))
h4 : ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), ∀ z ∈ Metric.ball s v, ‖F' z t‖ ≤ bound t
j : ℝ
hj : b < j
hj' : j < a
w : ℝ
hw1 : b < w
hw2 : w < j
w' : ℝ
hw1' : j < w'
hw2' : w' < a
t : ℝ
⊢ ‖log t • f t‖ = |log t| * ‖f t‖

Next tactic:
simp only [norm_smul, Real.norm_eq_abs]