Let \({\mathcal P}_n\) be the class of algebraic polynomials \(P(x)= \sum^n_{k= 0}a_kx^k\) of degree at most \(n\). The first inequality of the form \(| a_k|\leq C_{n,k}\| P\|\) was given by V. A. Markov in 1892. Then many mathematicians like I. Schur, Q. I. Rahman and G. Schmeisser improved the above inequality. Recently, G. V. Milovanović and A. Guessab have extended these results to polynomials with real coefficients which have \(m\) zeros on the real line.
In this paper, the author considers a more general problem including \(L^2\)-norm of polynomials with respect to a nonnegative measure on the real line \(\mathbb{R}\) and he gives the estimates for all coefficients. He also considers extremal problems for polynomials with respect to an inner product defined on the radial rays in the complex plane.
For the entire collection see [Zbl 0864.00057].