In this paper, we generalize a standard inequality for the Hardy space H-1 to involve a family of rational basis systems with cyclically repeated poles. This inequality is used to derive an inequality that gives information on the Fourier coefficients of functions in H-p (1 < p < 2) with respect to this family of basis functions. In an identification example, worst-case error of a least-squares algorithm for impulse input signal and this particular family of basis systems is computed.