INEQUALITIES FOR JACOBI POLYNOMIALS

Paul Turan observed that the Legendre polynomials satisfy the inequality Pn(x)2 − Pn−1(x)Pn+1(x)> 0,−1 ≤ x ≤ 1. And G. Gasper(ref. [6], ref. [7]) proved such an inequality for Ja-cobi polynomials and J. Bustoz and N. Savage (ref. [2]) proved Pαn (x)P β n+1(x) − Pαn+1(x)P βn (x)> 0, 12 ≤ α < β ≤ α + 2, 0 < x < 1, for the ultraspherical polynomials (respectively, Laguerre ploynomi-als). The Bustoz-Savage inequalities hold for Laguerre and ultras-pherical polynomials which are symmetric. In this paper, we prove some similar inequalities for non-symmetric Jacobi polynomials