On upper estimates of the partial sums of a trigonometric series in terms of lower estimates

Let $\{a_k\}_{k=0}^\infty$ and $\{b_k\}_{k=0}^\infty$ be sequences of real numbers and let $ S_n(x)$ be defined by
$$ S_n(x)=\sum^n_{k=0}\bigl(a_k\cos(kx)+b_k\sin(kx)\bigr),\qquad n=0,1,\dotsc\,. $$
It is proved that the estimate
$$ \max_x S_n(x)\leqslant 4a_0 n^{1-\alpha}, $$
holds for each natural number $n$ such that $S_m(x)\geqslant0$ for all $x$ and $m=1,\,\dots,\,n$. Here $\alpha\in(0,\,1)$ is the unique root of the equation
$$ \int^{3\pi /2}_0 t^{-\alpha}\cos t\,dt=0. $$
It is proved that the order $n^{1-\alpha}$ in this estimate cannot be improved. Various generalizations of this result are also obtained.