A note on sharp 1-dimensional Poincare inequalities

Let 1 < p < infinity and -infinity < a < b < infinity. We show by using elementary methods that the best constant C ( necessarily independent of a and b) for which the 1-dimensional Poincare inequality parallel to f - f(av)parallel to(1)(L)([a, b]) <= C(b - a)(2-1/p) parallel to f'parallel to L-P[a,L-b] holds for all Lipschitz continuous functions f, with f(av) = integral/(b-a), is C = 1/2(1+p')-1/p'.