| Title:
|
Further higher monotonicity properties of Sturm-Liouville functions (English) |
| Author:
|
Došlá, Zuzana |
| Author:
|
Háčik, Miloš |
| Author:
|
Muldoon, Martin E. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
29 |
| Issue:
|
1 |
| Year:
|
1993 |
| Pages:
|
83-96 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
n-times monotonic functions |
| Keyword:
|
completely monotonic functions |
| Keyword:
|
ultimately monotonic functions and sequences |
| Keyword:
|
regularly varying functions |
| Keyword:
|
Appell differential equation |
| Keyword:
|
generalized Airy equation |
| Keyword:
|
higher differences |
| MSC:
|
34B30 |
| MSC:
|
34C10 |
| MSC:
|
34D05 |
| idZBL:
|
Zbl 0812.34010 |
| idMR:
|
MR1242631 |
| . |
| Date available:
|
2008-06-06T21:24:00Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107469 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
[9] Lorch, L., Szego, P.: Higher monotonicity properties of certain Sturm-Liouville functions.Acta Math. 109 (1963), 55-73. MR 0147695 |
| Reference:
|
[10] Lorch, L., Muldoon, M. E., Szego, P.: Higher monotonicity properties of certain Sturm-Liouville functions. III.Canad. J. Math. 22 (1970), 1238-1265. MR 0274845 |
| Reference:
|
[11] Muldoon, M. E.: Higher monotonicity properties of certain Sturm-Liouville functions, V.Proc. Roy. Soc. Edinburgh 77A (1977), 23-37. Zbl 0361.34027, MR 0445033 |
| Reference:
|
[12] Seneta, E.: Regularly varying functions.Lecture Notes in Math., no. 508, Springer, 1976. Zbl 0324.26002, MR 0453936 |
| Reference:
|
[13] Vosmanský, J.: Monotonicity properties of zeros of the differential equation $y {^{\prime \prime }} + q(x)y = 0$.Arch. Math. (Brno) 6 (1970), 37-74. MR 0296420 |
| Reference:
|
[14] Williamson, R. E.: Multiply monotone functions and their Laplace transforms.Duke Math. J. 23 (1956), 189-207. Zbl 0070.28501, MR 0077581 |
| . |
