Generalization of an elementary inequality in Fourier analysis

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