A Schwarz lemma for two families of domains and complex geometry.

We make sharp estimates to obtain a Schwarz-type lemma for the symmetrized polydisc G n and for the extended symmetrized polydisc G ~ n . We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in [Young NJ. The automorphism group of the tetrablock. J Lond Math Soc. 2008;77(2):757–770]. Also we find a few geometric interplay between the members of the family G ~ n and its closure Γ ~ n . [ABSTRACT FROM AUTHOR]