Sharp Constant in Jackson's Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions

It is proved that, in the space $\mathrm{C}_{2\pi}$, for all $k,n\in\mathbb N$, $n>1$, the following inequalities hold:
$$ \biggl(1-\frac {1}{2n}\biggr)\frac{k^2+1}{2}\le \sup_{\substack{f\in \mathrm{C}_{2\pi}\\ f\ne\mathrm{const}}} \frac{{e}_{n-1}(f)}{\omega_2(f,\pi/(2nk))}\le \frac{k^2+1}{2}\mspace{2mu}. $$
where ${e}_{n-1}(f)$ is the value of the best approximation of $f$ by trigonometric polynomials and $\omega_2(f,h)$ is the modulus of smoothness of $f$. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.