The paper uses an idea of \textit{A. Ronveaux} [Math. Mag. 41, 231-234 (1968; Zbl 0203.393)] who observed that the logarithmic derivative of the solution y(x) of a second order differential equation satisfies a first order Riccati equation and that this leads to bounds on \(y'/y\) (and by integration this also gives bounds for y(x)). The authors work out this idea in more details and apply it to Bessel's differential equation to obtain upper and lower bounds for the ratio \(J_{\nu}(x)/J_{\nu}(y)\) of Bessel functions of the first kind and the ratio \(I_{\nu}(x)/I_{\nu}(y)\) of modified Bessel functions of the first kind.