Article

 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: [1] Appell, P.: Sur les transformations des équations différentielles linéaires.C. R. Acad. Sci. Paris 91 (1880), 211-214. Reference: [2] Borůvka, O.: Lineare Differentialtransformationen 2. Ordnung.VEB Verlag, Berlin, 1967, (English Translation, English Universities Press, London, 1973). Reference: [3] Došlá, Z.: Higher monotonicity properties of special functions: application on Bessel case $|\nu | < 1/2$.Comment. Math. Univ. Carolinae 31 (1990), 233-241. MR 1077894 Reference: [4] de Haan, L.: On regular variation and its application to the weak convergence of sample extremes.Mathematical Centre Tracts, vol. 32, Mathematisch Centrum, Amsterdam, 1975. Reference: [5] Feller, W.: An introduction to probability theory and its applications.vol. 2, 2nd ed., Wiley, 1971. Zbl 0219.60003 Reference: [6] Hartman, P.: On differential equations and the function $J_\nu ^2 + Y_\nu ^2$.Amer. J. Math. 83 (1961), 154-188. MR 0123039 Reference: [7] Hartman, P.: On differential equations, Volterra equations and the function $J_\nu ^2 + Y_\nu ^2$.Amer. J. Math. 95 (1973), 553-593. MR 0333308 Reference: [8] de La Vallée Poussin, Ch.-J.: Cours d’analyse infinitésimale.tome 1 , 12th ed, Louvain and Paris, 1959. 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 .

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