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As a special class of conformal metrics with negative curvatures, SK-metricsplay a crucial role in metric spaces. This paper concerns the variants ofAhlfors' lemma in an SK-metric space and gives a higher order derivativeformula for the logarithmic potential function, which can be applied for theestimates near the singularity of a conformal metric with negative curvatures.

Variants of Ahlfors' lemma and properties of the logarithmic potentials