On constants in the Jackson–Stechkin theorem in the case of approximation by algebraic polynomials

New estimates are proved for the constants $J(k,\alpha )$ in the classical Jackson–Stechkin inequality $E_{n-1}(f) \le J(k, \alpha ) \omega _k (f,{\alpha \pi }/{n})$, $\alpha >0$, in the case of approximation of functions $f \in C[-1,1]$ by algebraic polynomials. The main result of the paper implies the following two-sided estimates for the constants: $1/2\le J(2k,\alpha )<10$, $n \ge 2k(2k-1)$, $\alpha \ge 2$.