Further Higher Monotonicity Properties of Sturm{liouville Function

Suppose that the function q(t) in the diierential equation (1) y 00 + q(t)y = 0 is decreasing on (b;1) where b 0. We give conditions on q which ensure that (1) has a pair of solutions y 1 (t); y 2 (t) such that the n-th derivative (n 1) of the function p(t) = y 2 1 (t) + y 2 2 (t) has the sign (?1) n+1 for suuciently large t and that the higher diierences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.