Further higher monotonicity properties of Sturm-Liouville functions

Došlá, Zuzana; Háčik, Miloš; Muldoon, Martin E.

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Further higher monotonicity properties of Sturm-Liouville functions

Došlá, Zuzana ; Háčik, Miloš ; Muldoon, Martin E.

Archivum mathematicum, Tome 29 (1993) no. 1-2, pp. 83-96.

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

Résumé

Suppose that the function

q (t)

in the differential equation (1)

y^{''} + q (t) y = 0

is decreasing on

(b, \infty)

where

b \geq 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 \geq 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.

MR Zbl

Classification : 34B30, 34C10, 34D05
Mots-clés : 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},
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     number = {1-2},
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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. https://geodesic-test.mathdoc.fr/item/ARM_1993__29_1-2_a10/