The local uniform convergence of positive harmonic function sequence

Abstract

The Harnack distance on space  and its conformal invariance were constructed and studied by Herron. In this paper, we obtain the Harnack distance on domains  in . Then, we use this concept to investigate some properties of the positive harmonic function class. These results are obtained in the complex plane, so it is advantageous to take some tools of the complex analysis. The main result of this paper is the property of the local uniform convergence of the positive harmonic sequences on a domain in the complex plane.