English
Let f be concave on s and g be convex on s with f(x) ≤ 0 and g(x) ≥ 0 for x ∈ s. If hfg MonovaryOn f g s, then the product h = f • g is convex on s.
Русский
Пусть f конкавна, g выпукла на s, f(x) ≤ 0, g(x) ≥ 0; при условии MonovaryOn(f,g,s) имеем, что произведение $h=f•g$ выпуклое на s.
LaTeX
$$$\begin{aligned} &\text{ConcaveOn}(s,f) \land \text{ConvexOn}(s,g) \land \forall x\in s: f(x) \le 0, \ 0 \le g(x) \\ &\ &\text{MonovaryOn}(f,g,s) \Rightarrow \text{ConvexOn}(s, f\cdot g). \end{aligned}$$$
Lean4
theorem smul_convexOn' (hf : ConcaveOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) (hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0)
(hg₀ : ∀ ⦃x⦄, x ∈ s → 0 ≤ g x) (hfg : AntivaryOn f g s) : ConcaveOn 𝕜 s (f • g) :=
by
rw [← neg_convexOn_iff, ← smul_neg]
exact hf.smul'' hg.neg hf₀ (fun x hx ↦ neg_nonpos.2 <| hg₀ hx) hfg.neg_right
