Theorems · Theorem · real analysis

concaveOn_of_slope_anti_adjacent

∀ {𝕜 : Type u_1} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜},
  Convex 𝕜 s →
    (∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) → ConcaveOn 𝕜 s f

If for any three points x < y < z, the slope of the secant line of f : 𝕜 → 𝕜 on [x, y] is greater than the slope of the secant line of f on [y, z], then f is concave.

Defined in

Mathlib.Analysis.Convex.Slope

Cited by

1 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRing