Harmonic average of slopes and the stability of absolutely continuous invariant measure

For families of piecewise expanding maps which converge to a map with a fixed or periodic turning point touching a branch with slope of modulus equal to or less than 2, the standard Lasota-Yorke argument fails to prove stability. It is the goal of this paper to use instead the harmonic average of slopes condition for a large class of maps satisfying the summable oscillation condition for the reciprocal of the derivative. Using Rychlik's Theorem for a family of perturbations we prove weak compactness in L1 of the density functions associated with them. From this it follows that we have stability of absolutely continuous invariant measures of the limit map.