The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations

Abstract

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation1.1where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function1.2where y1(z) and y2(z) are two linearly independent solutions of1.3is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.