In this work, we study time evolution properties of univalent functions in the context of Hele–Shaw flows. We introduce the notion of circular tolerance and its generalization, the circular tolerance of order \(\beta \) , as new measures of distortion. We also define the concepts of restricted starlikeness and restricted strongly starlikeness. We investigate the time invariance of the properties of restricted starlikeness and restricted strongly starlikeness, both for the interior version of the Hele–Shaw problem. Our results show that, in the case of starlike dynamics and strongly starlike dynamics of order \(\beta \) , the upper bounds for circular tolerance and, respectively, for circular tolerance of order \(\beta \) are preserved, provided that these bounds are at least 1, for the inner version of the Hele–Shaw problem with injection and without surface tension. Several consequences of these results improve and refine previously known results on invariant geometric properties in Hele–Shaw dynamics.
