Generalization of an elementary inequality in Fourier analysis

The inequality $mathop {sup }limits_{n geqslant 1} left| {sumlimits_{k = 1}^n {frac{{sin kx}}{k}} } right| leqslant 3sqrt pi ,$mathop {sup }limits_{n geqslant 1} left| {sumlimits_{k = 1}^n {frac{{sin kx}}{k}} } right| leqslant 3sqrt pi , 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).