The Sturm-Liouville functions considered in this instalment are real (as are all other quantities discussed here) non-trivial solutions of the differential equation
1.1![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00037469/resource/name/S0008414X00037469_inline01.gif?pub-status=live)
Higher monotonicity properties, as defined in § 2, are investigated for a number of sequences (finite or infinite) associated with these functions. One such sequence, discussed in detail later, has the kth term
1.2![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00037469/resource/name/S0008414X00037469_inline02.gif?pub-status=live)
where the constant X > — 1 (to assure convergence of each integral), W(x) possesses higher monotonicity properties and, moreover, is such that, again, each integral converges, and X1, X2, … is a sequence (finite or infinite) of consecutive zeros of a solution of (1.1), which may or may not be linearly independent of y(x), in the interval of definition of the functions under consideration.