L¹-Convergence of the sine Fourier series with coefficients monotonic with respect to strongly regularly varying sequences

Let L¹(0,π) be the Banach space of all integrable functions on (0,π) with the usual L¹-norm. Let f ∈ L¹(0,π) be an odd function. We will show that, if the coefficients b_n of the sine Fourier series are strongly O-regularly varying quasimonotonic, the partial sums S_n(f, x) and the C − 1 means σ_n(f, x) of the sine Fourier series are equiconvergent, if b_n log n tends to zero as n tends to infinity. A necessary and sufficient condition for L¹ convergence of the sine Fourier series will follow.