  Let x(t) be a non-constant T-periodic solution to the ordinary differential
equation x'= f(x) in a Banach space X where f is assumed to be Lipschitz
continuous with constant L. Then there exists a constant c such that T L >= c,
with c only depending on X. It is known that c >= 6 in any Banach space and
that c = 2{\pi} in any Hilbert space, but whereas the bound of c = 2 pi is
sharp in any Hilbert space, there exists only one known example of a Banach
space such that c = 6 is optimal. In this paper, we show that the inequality is
in fact strict in any strictly convex Banach space. Moreover, we improve the
lower bound for l^p(R^n) and L^p(M, {\mu}) for a range of p close to p = 2 by
using a form of Wirtinger's inequality for functions in W^{1,p}([0, T ], L^p(M,
{\mu})).
