On a relationship in the theory of Fourier series

In this paper we prove the validity of the inequality
$$ \sup\limits_n\int_{-\pi}^\pi\Bigl|\frac{f(0)}2+\sum_{k=1}^nf\bigl(\frac{k\pi}n\bigr)e^{ikt}\Bigr|\,dt\le C\sum_{m=0}^\infty\Bigl|\int_0^\pi f(t)e^{imt}\,dt\Bigr| $$
for an arbitrary continuous function ($C$ is an absolute constant). An inequality in the opposite sense was obtained by one of us earlier.