, Volume 7, Issue 3, pp 655-672

Quasihyperbolic Distance in Punctured Planes

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Abstract

We give an explicit formula of the quasihyperbolic distance from a point to a line in the once punctured plane and prove the geodesic is orthogonal to the line. By this result, we give an affirmative answer to the open problem in the case of twice and thrice punctured planes raised by Klén and generalize their estimates. We also construct an example to show that the cosine inequality does not hold in twice or thrice punctured planes.

Communicated by Irene Sabadini.
This paper is supported by NNSF of China (11101165), Huaqiao University Fund (JB-ZR1136) and Macao Science and Technology Fund (FDCT/056/2010/A3).