Inequalities of Rafalson type for algebraic polynomials Original Research Article

For a positive Borel measure dμ, we prove that the constantγn(dν;dμ)≔supπ∈Pn⧹{0} ∫−∞∞π2(x)dν(x)∫−∞∞π2(x)dμ(x),can be represented by the zeros of orthogonal polynomials corresponding to dμ in case (i) dν(x)=(A+Bx)dμ(x), where A+Bx is nonnegative on the support of dμ and (ii) dν(x)=(A+Bx2)dμ(x), where dμ is symmetric and A+Bx2 is nonnegative on the support of dμ. The extremal polynomials attaining the constant are obtained and some concrete examples are given including Markov-type inequality when dμ is a measure for Jacobi polynomials.