English
Let f and g be concave on s, with f(x) ≤ 0, g(x) ≤ 0 on s, and MonovaryOn f g s. Then the product h = f • g is convex on s.
Русский
Пусть f и g конкавны на s, f(x) ≤ 0, g(x) ≤ 0 на s, и MonovaryOn(f,g,s). Тогда h=f•g выпукло на s.
LaTeX
$$$\begin{aligned} &\text{ConcaveOn}(s,f) \land \text{ConcaveOn}(s,g) \land \forall x\in s: f(x) \le 0, \ g(x) \le 0 \\ &\ &\text{MonovaryOn}(f,g,s) \Rightarrow \text{ConvexOn}(s, f\cdot g). \end{aligned}$$$
Lean4
theorem mul (hf : ConcaveOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x) (hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x)
(hfg : AntivaryOn f g s) : ConcaveOn 𝕜 s (f * g) :=
hf.smul' hg hf₀ hg₀ hfg
