  Let $L/K$ be an extension of complete discrete valuation fields of positive
characteristic, and assume that the residue field of $K$ is perfect. The
residue field of $L$ is not assumed to be perfect.
  In this paper, we show that the generalized Hasse-Herbrand function
$\psi_{L/K}^{\mathrm{ab}}$ has properties similar to those of its classical
counterpart. In particular, we prove that $\psi_{L/K}^{\mathrm{ab}}$ is
continuous, piecewise linear, increasing, convex, and satisfies certain
integrality properties.
