In this paper we deal with the number of zeros of a solution of the nth order linear differential equation
1.1![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00041511/resource/name/S0008414X00041511_eqn1.gif?pub-status=live)
where 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 function
1.2![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00041511/resource/name/S0008414X00041511_eqn2.gif?pub-status=live)
where y1(z) and y2(z) are two linearly independent solutions of
1.3![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00041511/resource/name/S0008414X00041511_eqn3.gif?pub-status=live)
is 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.