Results Math 72 (2017), 47–69
2017 Springer International Publishing
published online January 6, 2017
Results in Mathematics
Hyperbolic Type Distances in Starlike
Abstract. We study the growth of hyperbolic type distances in starlike
domains. We derive estimates for various hyperbolic type distances and
consider the asymptotic sharpness of the estimates.
Mathematics Subject Classiﬁcation. 30F45, 51M10, 30C65.
Keywords. Hyperbolic type distance, starlike domain.
The hyperbolic distance has turned out to be a useful tool in geometric func-
tion theory. The basic models for the hyperbolic distance are the unit ball
model and the upper half space model. Using these models in the plane case
n = 2, we can ﬁnd the hyperbolic distance in any simply connected domain
with at least 2 boundary points via the Riemann mapping theorem. In higher
dimensions n ≥ 3, there are no such results we could use to consider the hyper-
bolic distance in general domains. A solution to this is to use other distance
functions, which approximate the hyperbolic distance and are easier to eval-
uate. We call a distance function hyperbolic type, if it is comparable to the
The hyperbolic distance have had many applications to the geometric
function theory in the complex plain . The hyperbolic type metrics have
similar applications to geometric function theory in higher dimensions [14,15,
The study of hyperbolic type distances was initiated four decades ago by
Gehring and Osgood , Gehring and Palka  and Martin and Osgood .