English
In the complex setting, logDeriv of an infinite product equals the sum of logDeriv of factors, under summability and differentiability hypotheses.
Русский
В комплексном поле логарифмическая производная бесконечного произведения равна сумме логарифмических производных факторов при выполнении условий суммируемости и дифференцируемости.
LaTeX
$$$\displaystyle \logDeriv\Bigl(\prod_{i}^{\prime} f_i\cdot\Bigr)(x) = \sum_{i}^{\prime} \logDeriv\bigl(f_i\bigr)(x)$$$
Lean4
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has derivative `B'` everywhere on `ℝ`;
* `f` has right derivative `f'` at every point of `[a, b)`;
* we have `f' x < B' x` whenever `f x = B x`.
Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b))
(hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a)
(hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt)
(fun x _ => (hB x).hasDerivWithinAt) bound
