Geodesic
Positive Harmonic Functions and Complete Metrics
Canadian mathematical bulletin, Tome 32 (1989) no. 3, pp. 286-297

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DOI

We introduce the class of Harnack domains in which a Harnack type inequality holds for positive harmonic functions with bounds given in terms of the distance to the domain's boundary. We give conditions connecting Harnack domains with several different complete metrics. We characterize the simply connected plane domains which are Harnack and discuss associated topics. We extend classical results to Harnack domains and give applications concerning the rate of growth of various functions defined in Harnack domains. We present a perhaps new characterization for quasidisks. 
Herron, David A.; Schiff, Joel L. Positive Harmonic Functions and Complete Metrics. Canadian mathematical bulletin, Tome 32 (1989) no. 3, pp. 286-297. doi: 10.4153/CMB-1989-042-5
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     title = {Positive {Harmonic} {Functions} and {Complete} {Metrics}},
     journal = {Canadian mathematical bulletin},
     pages = {286--297},
     year = {1989},
     volume = {32},
     number = {3},
     doi = {10.4153/CMB-1989-042-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1989-042-5/}
}
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