Part Metric

A Metric defined by

$$d(z,w)=\sup\left[{\left\vert{\ln u(z)\over u(w)}\right\vert: u\in H^+}\right],$$

where $H^+$ denotes the Positive Harmonic Functions on a Domain. The part metric is invariant under Conformal Maps for any Domain.

References

Bear, H. S. ``Part Metric and Hyperbolic Metric.'' Amer. Math. Monthly 98, 109-123, 1991.