The following inequalities of Wirtinger and Poincaré establish a relationship between the norm of a function and that of its derivative.
(b) For any compact interval [a, b] and any continuously differentiable function f with f(a) = f(b) = 0, show that ∫(a to b) (f'(t))^2 dt < π^2 ∫(a to b) (f(t))^2 dt. Discuss the case of equality; and prove that the constant (b - a)^2 / π^2 cannot be improved. [Hint: Extend f to be odd with respect to a and periodic of period T = 2(b - a) so that its integral over an interval of length T is 0. Apply part (a) to get the inequality; and conclude that equality holds if and only if f(t) = Asin(πt)]