SOME MARKOV-BERNSTEIN TYPE INEQUALITIES AND CERTAIN CLASS OF SOBOLEV POLYNOMIALS

Let (μ0, μ1) be a vector of non-negative measures on the real line, with μ0 not identically zero, finite moments of all orders, compact or non compact supports, and at least one of them having an infinite number of points on its support. We show that for any linear operator T on the space of polynomials with complex coefficients and any integer n ≥ 0, there is a constant γn(T ) ≥ 0, such that ║Tp║S ≤ γn(T )║p║S, for any polynomial p of degree ≤ n, where γn(T ) is independent of p, and ║p║S ={|p(x)|2dμ0(x) +∫|p(x)|^2dμ1(x)}^1/2. We find a formula for the best possible value n(T) of γn(T ) and inequalities for n(T). Also, we give some examples when T is a differentiation operator and (μ0, μ1) is a vector of orthogonalizing measures for classical orthogonal polynomials.