TURAN INEQUALITIES AND ZEROS OF ORTHOGONAL POLYNOMIALS

We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence pk+1=xpk-ckpk-1, with a nondecreasing sequence {ck}. As a special case they include a non-asymptotic version of Mate, Nevai and Totik result on the largest zeros of orthogonal polynomials with ck=ck(2 delta)(1+o(k(-2/3))).our proof is based on new Turan inequalituies Which are obtained by analogy with higher order laguerre in equalities.