  In this paper we consider the Cauchy problem for higher order weakly
hyperbolic equations. We assume that the principal symbol depends only on one
space variable and the characteristic roots $\tau_j$ verify the inequality
\[\tau_j^2(x) + \tau_k^2(x) \le M \bigl(\tau_j(x)-\tau_k(x)\bigr)^2\] for some
constant $M$ independent of $x$. We prove that if the lower order terms verify
a suitable Levi condition, the Cauchy problem is well-posed in
$\mathcal{C}$-infinity.
