Some Remarks on the Green Function and the Azukawa Pseudometric

We show equality between the pluricomplex Green function (respectively, the Azukawa pseudometric) and the singular Carathéodory function (respectively, the singular Carathéodory pseudometric) for a wide class of planar domains. We also give a description of the Azukawa pseudometric of any domain in ℂn in the sense of Poletsky. All extremal discs for the Green function of any planar domain are also found. They are expressed in terms of its universal cover and Blaschke products.