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