Geodesic
Further higher monotonicity properties of Sturm-Liouville functions
Archivum mathematicum, Tome 29 (1993) no. 1-2, pp. 83-96.

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Suppose that the function $q(t)$ in the differential equation (1) $y^{\prime \prime }+q(t)y=0 $ is decreasing on $(b,\infty )$ where $b \ge 0$. 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 ($n\ge 1$) of the function $p(t)= y_1^2(t) +y_2^2(t)$ has the sign $(- 1)^{n+1}$ for sufficiently large $t$ and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.
Classification : 34B30, 34C10, 34D05
Keywords: n-times monotonic functions; completely monotonic functions; ultimately monotonic functions and sequences; regularly varying functions; Appell differential equation; generalized Airy equation; higher differences 
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     author = {Do\v{s}l\'a, Zuzana and H\'a\v{c}ik, Milo\v{s} and Muldoon, Martin E.},
     title = {Further higher monotonicity properties of {Sturm-Liouville} functions},
     journal = {Archivum mathematicum},
     pages = {83--96},
     publisher = {mathdoc},
     volume = {29},
     number = {1-2},
     year = {1993},
     mrnumber = {1242631},
     zbl = {0812.34010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1993__29_1-2_a10/}
}
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Došlá, Zuzana; Háčik, Miloš; Muldoon, Martin E. Further higher monotonicity properties of Sturm-Liouville functions. Archivum mathematicum, Tome 29 (1993) no. 1-2, pp. 83-96. http://geodesic.mathdoc.fr/item/ARM_1993__29_1-2_a10/