Generalized Hasse-Herbrand functions in positive characteristic

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.