Article

Full entry | PDF   (0.2 MB)
Keywords:
n-times monotonic functions; completely monotonic functions; ultimately monotonic functions and sequences; regularly varying functions; Appell differential equation; generalized Airy equation; higher differences
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.
References:
[1] Appell, P.: Sur les transformations des équations différentielles linéaires. C. R. Acad. Sci. Paris 91 (1880), 211-214.
[2] Borůvka, O.: Lineare Differentialtransformationen 2. Ordnung. VEB Verlag, Berlin, 1967, (English Translation, English Universities Press, London, 1973).
[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
[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.
[5] Feller, W.: An introduction to probability theory and its applications. vol. 2, 2nd ed., Wiley, 1971. Zbl 0219.60003
[6] Hartman, P.: On differential equations and the function $J_\nu ^2 + Y_\nu ^2$. Amer. J. Math. 83 (1961), 154-188. MR 0123039
[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
[8] de La Vallée Poussin, Ch.-J.: Cours d’analyse infinitésimale. tome 1 , 12th ed, Louvain and Paris, 1959.
[9] Lorch, L., Szego, P.: Higher monotonicity properties of certain Sturm-Liouville functions. Acta Math. 109 (1963), 55-73. MR 0147695
[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
[11] Muldoon, M. E.: Higher monotonicity properties of certain Sturm-Liouville functions, V. Proc. Roy. Soc. Edinburgh 77A (1977), 23-37. MR 0445033 | Zbl 0361.34027
[12] Seneta, E.: Regularly varying functions. Lecture Notes in Math., no. 508, Springer, 1976. MR 0453936 | Zbl 0324.26002
[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
[14] Williamson, R. E.: Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 (1956), 189-207. MR 0077581 | Zbl 0070.28501

Partner of