On bounds for solutions of monotonic first order difference-differential systems - Segura, Javier (Journal of inequalities and applications, 2012)
Many special functions are solutions of first order linear systems y n ′ ( x ) = a n ( x ) y n ( x ) + d n ( x ) y n - 1 ( x ) , y n - 1 ′ ( x ) , = b n ( x ) y n - 1 ( x ) + e n ( x ) y n ( x ) . We obtain bounds for the ratios y n ( x )/ y n -1 ( x ) and the logarithmic derivatives of y n ( x ) for solutions of monotonic systems satisfying certain initial conditions. For the case d n ( x ) e n ( 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 d n ( x ) e n ( 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). Mathematics Subject Classification 2000 : 33CXX; 26D20; 34C11; 34C10; 39A06.