Summary: The classical Markov (1889) and Bernstein (1912) inequalities and corresponding extremal problems were generalized for various domains, various norms and for various subclasses of polynomials, both algebraic and trigonometric. Besides some classical results in uniform norm, we give a short account of \(L^r\) inequalities of Markov type for algebraic polynomials, with a special attention to the case \(r=2\). We also study extremal problems of Markov’s type \[ C_{n,m}= \sup_{P\in{\mathcal P}_n} {\|{\mathcal D}_m P\|\over\|A^{m/2}P\|}, \] where \({\mathcal P}_n\) is the class of all algebraic polynomials of degree at most \(n\), \(d\lambda(t)= w(t) dt\) is a nonnegative measure corresponding to the classical orthogonal polynomials, \(A\in{\mathcal P}_2\), \(\|P\|= (\int_{\mathbb{R}}|P(t)|^2d\lambda(t))^{1/2}\), and \({\mathcal D}_m\) is a differential operator defined by \[ {\mathcal D}_m P= {d^m\over dt^m} [A^mP]\qquad (P\in{\mathcal P}_n,\;m\geq 1). \]
For the entire collection see [Zbl 0947.00027].