/-- Log-convexity of the Gamma function on the positive reals (stated in multiplicative form),
proved using the Hölder inequality applied to Euler's integral. -/
theorem Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma {s t a b : ℝ} (hs : 0 < s) (ht : 0 < t)
    (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
    Gamma (a * s + b * t) ≤ Gamma s ^ a * Gamma t ^ b := by
  -- We will apply Hölder's inequality, for the conjugate exponents `p = 1 / a`
  -- and `q = 1 / b`, to the functions `f a s` and `f b t`, where `f` is as follows:
  let f : ℝ → ℝ → ℝ → ℝ := fun c u x => exp (-c * x) * x ^ (c * (u - 1))
  have e : IsConjugateExponent (1 / a) (1 / b) := Real.isConjugateExponent_one_div ha hb hab
  have hab' : b = 1 - a := by linarith
  have hst : 0 < a * s + b * t := add_pos (mul_pos ha hs) (mul_pos hb ht)
  -- some properties of f:
  have posf : ∀ c u x : ℝ, x ∈ Ioi (0 : ℝ) → 0 ≤ f c u x := fun c u x hx =>
    mul_nonneg (exp_pos _).le (rpow_pos_of_pos hx _).le
  have posf' : ∀ c u : ℝ, ∀ᵐ x : ℝ ∂volume.restrict (Ioi 0), 0 ≤ f c u x := fun c u =>
    (ae_restrict_iff' measurableSet_Ioi).mpr (ae_of_all _ (posf c u))
  have fpow :
    ∀ {c x : ℝ} (_ : 0 < c) (u : ℝ) (_ : 0 < x), exp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) := by
    intro c x hc u hx
    dsimp only
    rw [mul_rpow (exp_pos _).le ((rpow_nonneg_of_nonneg hx.le) _), ← exp_mul, ← rpow_mul hx.le]
    congr 2 <;> · field_simp [hc.ne']; ring
  -- show `f c u` is in `ℒp` for `p = 1/c`:
  have f_mem_Lp :
    ∀ {c u : ℝ} (hc : 0 < c) (hu : 0 < u),
      Memℒp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0)) := by
    intro c u hc hu
    have A : ENNReal.ofReal (1 / c) ≠ 0 := by
      rwa [Ne.def, ENNReal.ofReal_eq_zero, not_le, one_div_pos]
    have B : ENNReal.ofReal (1 / c) ≠ ∞ := ENNReal.ofReal_ne_top
    rw [← memℒp_norm_rpow_iff _ A B, ENNReal.toReal_ofReal (one_div_nonneg.mpr hc.le),
      ENNReal.div_self A B, memℒp_one_iff_integrable]
    · apply Integrable.congr (GammaIntegral_convergent hu)
      refine' eventuallyEq_of_mem (self_mem_ae_restrict measurableSet_Ioi) fun x hx => _
      dsimp only
      rw [fpow hc u hx]
      congr 1
      exact (norm_of_nonneg (posf _ _ x hx)).symm
    · refine' ContinuousOn.aestronglyMeasurable _ measurableSet_Ioi
      refine' (Continuous.continuousOn _).mul (ContinuousAt.continuousOn fun x hx => _)
      · exact continuous_exp.comp (continuous_const.mul continuous_id')
      · exact continuousAt_rpow_const _ _ (Or.inl (mem_Ioi.mp hx).ne')
  -- now apply Hölder:
  rw [Gamma_eq_integral hs, Gamma_eq_integral ht, Gamma_eq_integral hst]
  convert
    MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg e (posf' a s) (posf' b t) (f_mem_Lp ha hs)
      (f_mem_Lp hb ht) using
    1
  · refine' set_integral_congr measurableSet_Ioi fun x hx => _
    dsimp only
    have A : exp (-x) = exp (-a * x) * exp (-b * x) := by
      rw [← exp_add, ← add_mul, ← neg_add, hab, neg_one_mul]
    have B : x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1)) := by
      rw [← rpow_add hx, hab']; congr 1; ring
    rw [A, B]
    ring
  · rw [one_div_one_div, one_div_one_div]
    congr 2 <;> 

Proof state:
case h.e'_4.e_a.e_a
s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))
e : IsConjugateExponent (1 / a) (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂Measure.restrict volume (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
f_mem_Lp : ∀ {c u : ℝ}, 0 < c → 0 < u → Memℒp (f c u) (ENNReal.ofReal (1 / c))
⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (t - 1) = ∫ (a : ℝ) in Ioi 0, f b t a ^ (1 / b)

Next tactic:
exact set_integral_congr measurableSet_Ioi fun x hx => fpow (by assumption) _ hx