Abstract. We give a Sturm-type comparison theorem and a convexity theorem for difference equations. We apply the convexity results to discrete orthogonal polynomials, such as the Hahn and Meixner polynomials by obtaining estimates on the second difference of their zeros. We show that the corresponding theorems for $q$-difference equations also hold, and present the results on the $q$-Laguerre polynomials.

AMS Subject Classification (1991): 33C45, 33D45, 39A12, 39A13

Keyword(s): Sturm comparison theorem, Sturm convexity theorem, self-adjoint equations, orthogonal polynomials, $q$-difference equations, convexity of zeros

Received January 10, 2012, and in revised form April 2, 2012. (Registered under 3/2012.)