@article{10902/579,
year = {2012},
url = {http://hdl.handle.net/10902/579},
abstract = {Many special functions are solutions of first order linear systems
y_ n(x) = an(x)yn(x) + dn(x)yn−1(x), y_n−1(x), = bn(x)yn−1(x) + en(x)yn(x) . We obtain
bounds for the ratios yn(x)/yn-1(x) and the logarithmic derivatives of yn(x) for solutions
of monotonic systems satisfying certain initial conditions. For the case dn(x)en(x) > 0,
sequences of upper and lower bounds can be obtained by iterating the recurrence
relation; for minimal solutions of the recurrence these are convergent sequences. The
bounds are related to the Liouville-Green approximation for the associated second
order ODEs as well as to the asymptotic behavior of the associated three-term
recurrence relation as n ® +∞; the bounds are sharp both as a function of n and x.
Many special functions are amenable to this analysis, and we give several examples
of application: modified Bessel functions, parabolic cylinder functions, Legendre
functions of imaginary variable and Laguerre functions. New Turán-type inequalities
are established from the function ratio bounds. Bounds for monotonic systems with
dn(x)en(x) < 0 are also given, in particular for Hermite and Laguerre polynomials of
real positive variable; in that case the bounds can be used for bounding the
monotonic region (and then the extreme zeros).},
organization = {This study was supported by the Ministerio de Ciencia e Innovación, project MTM2009-11686. The author thanks the two anonymous referees for helpful comments.},
publisher = {SpringerOpen},
publisher = {Journal of Inequalities and Applications, 2012, 65},
title = {On bounds for solutions of monotonic first order difference-differential systems},
author = {Segura Sala, José Javier},
}