  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 $n\rightarrow
+\infty$; 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 Tur\'an-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).
