The exact order of approximation of functions by Bernstein polynomials in a Hausdorff metric

We investigate the approximation of functions by Bernstein polynomials. We prove that
$$ r_{[0,1]}(f, B_n(f))\leqslant\mu_f\left(4\sqrt{\frac{\ln n}{n}}\right)+O\left(\sqrt{\frac{\ln n}{n}}\right),\eqno{(1)} $$
where $r_{[0,1]}(f, B_n(f))$ is the Hausdorff distance between the functions $f(x)$ and $B_n(f; x)$ in $[0,1]$,
$$ \mu_f(\delta)=\frac12\sup_{\substack{|x_1-x_2|\leqslant\delta\\ x_1,x_2\in\Delta}}\{\sup_{x_1\leqslant x\leqslant x_2}[|f(x_1)-f(x)|+|f(x_2)-f(x)|]-|f(x_1)-f(x_2)|\} $$
is the modulus of nonmonotonicity of $f(x)$. The bound (1) is of better order than that obtained by Sendov. We show that the order of (1) cannot be improved.