  For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval
$(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is
regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl
coefficient. De Branges' inverse spectral theorem states that the assignment
$H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and
Nevanlinna functions.
  We give upper and lower bounds for $|q_H(z)|$ and $\operatorname{Im} q_H(z)$
when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the
Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to
universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is
independent of the off-diagonal entries of $H$ and depends monotonically on the
diagonal entries in a natural way. The imaginary part is, in general, not fully
determined by our bounds (in forthcoming work we shall prove that for "most"
Hamiltonians also $\operatorname{Im} q_H(z)$ is fully determined).
  We translate the asymptotic behaviour of $q_H$ to the behaviour of the
spectral measure $\mu_H$ of $H$ by means of Abelian--Tauberian results and
obtain conditions for membership of growth classes defined by weighted
integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$
w.r.t. a weight function. Moreover, we apply our results to Krein strings and
Sturm--Liouville equations.
