The paper deals with simultaneous approximation by the sequence of positive linear operators \(M_ n\) given by \[ (M_ nf)(x)=(n+1)\sum^ n_{k=0}p_{nk}(x)\int^ 1_ 0p_{nk}(t)f(t)dt, \] \(x\in[0,1]\), \(f\in L_ 1[0,1]\), \(p_{nk}(x)={n\choose k}x^ k(1-x)^{n-k}\). This modification of the Bernstein operators is due to Durrmeyer. The main result is the equivalence of \(\| (M_ nf-f)^{(s)}\|_ p\leq C_ f\left\{\omega\left({1\over\sqrt n}\right)+{1\over n}\right\}\) and \(\omega^ 2_ \varphi(f^{(s)},t)_ p\leq C_ f\{\omega(t)+t^ 2\}\) for \(f\in L^ s_ p[0,1]\), \(1\leq p<\infty\), \(s\in\mathbb{N}\). The increasing function \(\omega\) is assumed to satisfy \(0\leq\omega(kt)\leq Ak^ 2\omega(t)\) for \(t\leq 0\), \(k\in\mathbb{N}\). For instance one can choose \(\omega(t)=t^ \alpha|\log t|^ \beta\), \(0<\alpha\leq 2\), \(0\leq\beta<\infty\). The result supplements and extends an earlier direct theorem of Heilmann and Müller.