Generalization of an elementary inequality in Fourier analysis

The inequality sup(n≥1)|(n)∑(k=1) sin/kx/k|3≤(π)plays an important role in Fourier analysis and approximation theory. It has recently been generalized by Telyakovskii and Leindler. This paper further generalizes and improves their results by introducing a new class of sequences called γ-piecewise bounded variation sequence(γ-PBVS).