Coherent pairs of measures and Markov-Bernstein inequalities

All the coherent pairs of measures associated to linear functionals c0 and c1, introduced by Iserles et al. in 1991, have been given by Meijer in 1997. There exist seven kinds of coherent pairs. All these cases are explored in order to give three term recurrence relations satisfied by polynomials. The smallest zero μ1,n of each of them of degree n has a link with the Markov-Bernstein constant Mn appearing in the following Markov-Bernstein inequalities:c1((p′)2)≤Mn2c0(p2),∀p∈Pn, where Mn=1μ1,n. The seven kinds of three term recurrence relations are given. In the case where c0=e−xdx+δ(0) and c1=e−xdx, explicit upper and lower bounds are given for μ1,n, and the asymptotic behavior of the corresponding Markov-Bernstein constant is stated. Except in a part of one case, limn→∞⁡μ1,n=0 is proved in all the cases.