Sharp inequality of Jackson–Stechkin type and widths of functional classes in the space $L_2$

For classes of differentiable periodic functions, defined by means of generalized moduli of continuity $\Omega_m(f,t)$, satisfying the condition
$$ \left(\int_0^h\Omega_m^{2/m}(f^{(r)},t)dt\right)\leqslant\Phi(h), $$
where $m\in\mathbb{N}$, $r\in\mathbb{Z}_+$, $h>0$ and $\Phi$ is a given majorant, under certain restrictions on the majorant, the exact values of various $n$-widths in the space $L_2$ are calculated.