Problem of correctness of the best approximation in the space of continuous functions

Let $W^rH_\omega$ the subclass of those functions of $C^r[a,b]$, for which $\omega(f^{(r)},\delta)\le\omega(\delta)$, where $\omega(\delta)$ is a given modulus of continuity, and $P_n$ be the space of algebraic polynomials of degree at most $n$ and $\pi_n(f)$ be the polynomial of best approximation for $f(x)$ on $[a,b]$. Estimates for
$$ A_1(\varepsilon)=\sup_{f\in W^rH_\omega}\sup_{\substack{q_n\in P_n\\\|f-q_n\|\le\|f-\pi_n(f)\|+\varepsilon}}\|\pi_n(f)-q_n\|, $$
and moduli of continuity of the operators of best approximation on $W^rH_\omega$ are established. For example, if $\omega(\delta)=\delta^\alpha$, then
\begin{alignat*}{2} A_1(\varepsilon)&\asymp\varepsilon^{(r+\alpha)/(n+r+\alpha)}&&\quad\text{for }\varepsilon<1, \\ A_1(\varepsilon)&\asymp\varepsilon&&\quad\text{for }\varepsilon>1. \end{alignat*}