Many special functions are solutions of first order linear systems . 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.