Abstract: Many special functions are solutions of first order linear systems y?(x) = a(x)y(x) + d(x)w(x), w?(x) = b(x)w(x) + e(x)y(x). We obtain bounds for the logarithmic derivatives of the solutions of monotonic systems satisfying certain initial conditions. In particular, by Liouville-transforming the first order system equivalent to the second order ODE y??(x) + A(x)y(x) = 0 we obtain bounds related to the Liouville?Green approximation and find conditions under which such approximation is a bound for some solutions. We illustrate this with the Airy equation.

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