MARKOV-BERNSTEIN TYPE INEQUALITIIES FOR POLYNOMIALS

MARKOV-BERNSTEIN TYPE INEQUALITIIES FOR POLYNOMIALS

Let $\mu$(x) be an increasing function on the real line with finite moments of all oeders. We show that for any linear operator T on the space of polynomials and any interger n $\geq$ 0, there is a constant $\gamma n(T)\geq0$, independent of p(x), such that $\parallel T_p\parallel\leq\gamma n(T)\parallel P\parallel$, for any polynomial p(x) of degree $\leq$ n, where We find a formular for the best possible value $\Gamma_n(T)\;of\;\gamma n(T)$ and estimations for $\Gamma_n(T)$. We also give several illustrating examples when T is a differentiation or a difference operator and $d\mu$(x) is an orthogonalizing measure for classical or discrete orthogonal polynomials.