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

在李等人。（J Aust Math Soc 97：383–390，2014），作者讨论了Banach空间中的两类映射，它们是距离比度量中的\（\ varphi \）-拟等距映射（简称\（\ varphi \） -DRQI映射）和距离比率度量中的完全（（\ varphi \）-拟等距映射（简称\（\ varphi \）- FDRQI映射），其中\（\ varphi：\） \（[0，\ infty） \ rightarrow [0，\ infty）\）表示准等距控制功能。在本文中，我们通过引入一类新的映射（即\（\ varphi \）- PDRQI映射）来继续进行此讨论。本文的目的是双重的。让f是Banach空间中两个适当域之间的同胚。首先，我们证明f为\（\ varphi \）- PDRQI等效于f为\（\ varphi \）- FDRQI。然后，作为获得的等价关系的应用，我们证明f为\（\ varphi \）- DRQI，f为\（\ varphi \）- PDRQI和f为\（\ varphi \）- FDRQI等效于每个当准等距控制函数的形式为\（\ varphi（t）= Mt \）时。