For the class of Fourier serîes with 8-quasîmonotone coefficients, itiş proved tbat I i Sn-'Jn I I = 0(0. “ co , if and only if a^^ log n = o(l), n CO . This generalizes the theorem of Garrett, Rees and Stanojevic [3], and Telyakovskii and Fomine [6] for quasi-monotone, and monotone coefficients respectively. 1. A seguence {a„} of positive numbers is said. to be quasi- monotone if Aa, —.a — for some positive k, where Aaj, »n —' ^n+ı. It is obvious tbat every null monotonic decreasing sequence is quasi-monotone. The sequeııce {aj,} is said to be S- quasi-monotone if a^‘n o, a.n o ultimately and Aa^ > — wbere {S^} is a sequence of positive numbers. Clearly a null quasi- monotone sequence is S-quasi-monotone witb Sn= n 2. The problem of L-*convergence of Fourier cosine seri es f(x) = co + 2 n=ı 12 »n cos nx has been settied for various special class of coefficients, (See e.g. Young [7], Kolmogorov [4], Fomine [1], Garrett and Stano­ jevic [2], Telyakovskii and Fomine [6], ete). RecCntly, Garrett, Rees and Stanojevic [3] proved the fol­ îowing theorem which is too a generalization of a result of Telya- kovskii and Fcmine ([6], Theorem 1).