  For any real limit-$n$ $2n$th-order selfadjoint linear differential
expression on $[0,\infty)$, Titchmarsh- Weyl matrices
  $M(\lambda)$ can be defined. Two matrices of particu lar interest are the
matrices $M_D(\lambda)$ and $M_N(\lambda)$ assoc iated respectively with
Dirichlet and Neumann boundary conditions at $x=0$. These satisfy
  $M_D(\lambda) = -M_{N}(\lambda)^{-1}$. It is known that when these matrices
have poles (which can only lie on the real axis) the existence of valid HELP
inequalities depends on their behaviour in the neighbourhood of these poles. We
prove a conjecture of Bennewitz and use it, together with a new algorithm for
computing the Laurent expansion of a Titchmarsh-Weyl matrix in the
neighbourhood of a pole, to investigate the existence of HELP inequalities for
a number of differential equations which have so far proved awkward to analyse
