Theorems · Theorem · real analysis

ConvexOn.secant_mono

∀ {𝕜 : Type u_1} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜},
  ConvexOn 𝕜 s f →
    ∀ {a x y : 𝕜}, a ∈ s → x ∈ s → y ∈ s → x ≠ a → y ≠ a → x ≤ y → (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)

If f : 𝕜 → 𝕜 is convex, then for any point a the slope of the secant line of f through a and b ≠ a is monotone with respect to b.

Defined in

Mathlib.Analysis.Convex.Slope

Cited by

3 results in Mathlib

Foundations

Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRing