Barrlund's distance function and quasiconformal maps
Marcelina MocanuComplex Variables and Elliptic Equations
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Abstract
Answering a question about triangle inequality suggested by R. Li, A. Barrlund [2] introduced a distance function which is a metric on a subdomain of R n. We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps. z∈∂G |z 1 − z 2 | |z 1 − z| + |z − z 2 | , z 1 , z 2 ∈ G , was recently studied in [5, 11]. As shown in [11], this metric is closely related to the quasihyperbolic metric [10, 8, 24] and several other metrics of current interest [14, 20, 19, 9]. We study the Barrlund metric b G,p and compare it to s G = b G,1. For the cases of a ball or a half-plane we give in our main theorems 3.27 and 3.24 explicit formulas for b G,2. To this end, we first recall some properties of s G. By compactness, the suprema in (1.1) and (1.2) are attained. If G is convex, it is simple to see that the extremal point z 0 for (1.2) is a point of contact of the boundary with an ellipse contained in G with foci at z 1 , z 2. We prove the following sharp inequality between the above two metrics. Theorem 1.3. Let G be a domain in R n and let p ≥ 1. Then for all points z 1 , z 2 ∈ G s G (z 1 , z 2) ≤ b G,p (z 1 , z 2) ≤ 2 1−1/p s G (z 1 , z 2). Clearly, this inequality holds as an identity if p = 1. But perhaps more interesting is that the right hand side holds as an equality for all p ≥ 1 if G = {z ∈ C : Im (z) > 0} , and z 1 , z 2 ∈ G with Im(z 1) = Im(z 2) .
