Quasiisometric mappings in the distance ratio metric in Banach spaces

In Li et al. (J Aust Math Soc 97:383–390, 2014), the authors discussed two classes of mappings in Banach spaces, which are φ-quasiisometric mappings in the distance ratio metric (briefly, φ-DRQI mappings) and fully φ-quasiisometric mappings in the distance ratio metric (briefly, φ-FDRQI mappings), where φ: [0 , ∞) → [0 , ∞) denotes a quasiisometric control function. In this paper, we continue this discussion by introducing a new class of mappings, that is, φ-PDRQI mappings. The purpose of this paper is twofold. Let f be a homeomorphism between two proper domains in a Banach space. First, we prove that f being φ-PDRQI is equivalent to f being φ-FDRQI. Then, as an application of the obtained equivalent relation, we show that f being φ-DRQI, f being φ-PDRQI and f being φ-FDRQI are equivalent to each other when the quasiisometric control functions have the form φ(t) = Mt.