Assume a function v() with and v() = 0. a) Show (v())2 ( ) (v

Assume a function v(ζ) with α ≤ ζ ≤ β and v(α) = 0.

a) Show (v(ζ))2 ≤ (ζ − α)  ζ α (v (x))2 dx . (Hint: Recall v(ζ) = ζ α v (x) dx, and apply the Schwarzian inequality (C3.7).)

b) Use

a) to show  β α (v(ζ))2 dζ ≤ 1 2 (β − α) 2  β α (v (x))2 dx .