English
Let f be concave on s and g be convex on s with f(x) ≤ 0 and g(x) ≤ 0 for x ∈ s, and assume MonovaryOn f g s. Then the product h = f · g is convex on s.
Русский
Пусть f конкавна на s, g выпукла на s, f(x) ≤ 0, g(x) ≤ 0 на s, и MonovaryOn(f,g,s). Тогда h=f g выпукло на s.
LaTeX
$$$\begin{aligned} &\text{ConcaveOn}(s,f) \land \text{ConvexOn}(s,g) \land \forall x\in s: f(x) \le 0 \land g(x) \le 0 \\ &\ &\text{MonovaryOn}(f,g,s) \Rightarrow \text{ConvexOn}(s, f\cdot g). \end{aligned}$$$
Lean4
theorem smul_convexOn [IsOrderedModule 𝕜 E] (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ f x)
(hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : MonovaryOn f g s) : ConvexOn 𝕜 s (f • g) :=
by
rw [← neg_concaveOn_iff, ← smul_neg]
exact hf.smul' hg.neg hf₀ (fun x hx ↦ neg_nonneg.2 <| hg₀ hx) hfg.neg_right
