Some inequalities between the best polynomial approximation and averaged finite-difference norms in space $L_2$

Exact constants are found in inequalities type Jackson-Stechkin for smoothness characte-ristics $\Lambda_{m}(f), m \in\mathbb{N} $ determined by averaging the norm in $L_{2}$ of finite differences of the $m$-th order of the functions $f$. For function classes, defined by the smoothness characteristic $\Lambda_{m}(f)$, and the majorant $\Phi $ satisfying a certain condition, calculated the exact values of different $n$-widths.