On extreme zeros of classical orthogonal polynomials

Let x 1 and x k be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree k . We shall establish sharp inequalities of the form x 1 < A , x k > B , which are uniform in all the parameters involved. Together with inequalities in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with a high precision.