Let \(I\) be a closed real interval, a function \(\varphi\in C(I)\) is called a weight function if \(\varphi(x)>0\), \(x\in I^0\) (the interior set of \(I)\). Given a weight function \(\varphi\), let \(\omega^2_\varphi (f, \varepsilon)_\infty\) be the Ditzian-Totik second modulus of smoothness of \(f\in C(I)\). Let \(\varphi\) be a weight function and let \(\delta_\varphi>0\). The authors consider a family \(\mathbb{L}:=(L_\delta, 00\). (3) Bernstein polynomials, let \(f\in\mu (\varphi)\), with \(\varphi(x) =\sqrt{x (1-x)}\) \(x\in [0,1]\), then \(\|B_nf- f\|\leq 4\omega^2_\varphi (f;{1 \over \sqrt n})\), \(n=1,2, \dots\).