The paper contains far reaching extensions of the classical Bernstein and Markoff inequalities for the derivatives of polynomials. For the Bernstein case (pointwise estimate) the author shows the following result: Let \(E\) consist of a finite number of compact intervals and let \(\omega_E\) be the density of the equilibrium measure of \(E\). Then, for any positive integer \(n\) and any polynomial \(P\) of degree \(\leq n\), we have \[ \bigl|P'(x)\bigr |\leq\pi \omega_E(x) n\|P\|_E\;(x\in E) \tag{1} \] (where \(\|\cdot \|_E\) denotes the sup-norm on \(E)\). A similar result holds for arbitrary compact sets \(E\) on the real axis and for interior points \(x\) of \(E\) (if existent). Moreover, (1) turns out to be sharp in the sense that, for every \(\varepsilon >0\), every \(x\in\text{int}(E)\) and every \(n\) large enough, there is a polynomial \(P\not\equiv 0\) of degree \(\leq n\) such that \(|P'(x) |> (1-\varepsilon) \pi\omega_E (x)n\|P\|_E\). To formulate the extension of Markoff’s inequality for the finite union of intervals \(E=\cup^\ell_{i=1} [a_{2i -1}, a_{2i}]\), the author considers the sets \(E^j:= \{x\in E:|x-a_j |< |x-a_i |\), \(i\neq j\}\) of parts of \(E\) lying closer to \(a_j\) than to any other endpoint. The generalization of Markoff’s inequality turns out to be a set of inequalities, one around each endpoint \(a_j\): For each \(1\leq j\leq 2\ell\) we have \(\|P'\|_{E^j} \leq(1+o (1))M_jn^2 \|P\|_E\), with explicitly given (and asymptotically sharp) constants \(M_j\). The proofs use sets that are obtained as the inverse image of intervals under (special) polynomial mappings and an approximation result for general finite unions of compact intervals by such sets.