A note on the absolute convergence of lacunary Fourier series

P. B. Kennedy [3] studied lacunary Fourier series whose generating functions are of bounded variation on a subinterval I of [-g, g] and satisfy a Lispschitz condition of order a on I. We show that the conclusion of one of his theorems on the absolute convergence of Fourier series remains valid when the function is merely of bounded rth variation in I and belongs to a class Lip(a, p) in I. Our results also generalize three theorems of S. M. Mazhar [4].