Higher monotonicity properties of certain Sturm-Liouville functions. III

A Sturm-Liouville function is simply a non-trivial solution of the Sturm-Liouville differential equation (1.1) considered, together with everything else in this study, in the real domain. The associated quantities whose higher monotonicity properties are determined here are defined, for fixed λ > –1, to be (1.2) where y(x) is an arbitrary (non-trivial) solution of (1.1) and x 1, x 2, … is any finite or infinite sequence of consecutive zeros of any non-trivial solution z(x) of (1.1) which may or may not be linearly independent of y(x). The condition λ > –1 is required to assure convergence of the integral defining Mk , and the function W(x) is taken subject to the same restriction.