  Chaundy and Jolliffe [4] proved that if $\{a_{n}\}$ is a non-increasing
(monotonic) real sequence with $\lim\limits_{n\to \infty}a_{n}=0$, then a
necessary and sufficient condition for the uniform convergence of the series
$\sum_{n=1}^{\infty}a_{n}\sin nx$ is $ \lim\limits_{n\to \infty}na_{n}=0$. We
generalize (or weaken) the monotonic condition on the coefficient sequence
$\{a_{n}\}$ in this classical result to the so-called mean value bounded
variation condition and prove that the generalized condition cannot be weakened
further. We also establish an analogue to the generalized Chaundy and Jolliffe
theorem in the complex space.
