Schwarz lemma and Kobayashi metrics for holomorphic functions

In this paper, we mainly consider Schwarz lemma for holomorphic functions and contraction properties of holomorphic functions with respect to Kobayashi distances. We also review some results related to the subject using some novelty and announce a few new results.

Estimates of Invariant Metrics on Pseudoconvex Domains of Finite Type inC3

LetΩbe a smoothly bounded pseudoconvex domain inC3and assume thatz0∈bΩis a point of finite 1-type in the sense of D’Angelo. Then, there are an admissible curveΓ⊂Ω∪{z0}, connecting points q0∈Ωandz0∈bΩ, and a quantityM(z,X), alongz∈Γ, which bounds from above and below the Bergman, Caratheodory, and Kobayashi metrics in a small constant and large constant sense.

The asymptotic Teichmüller space and the asymptotic Grunsky map

The Grunsky map is known to be holomorphic on the universal Teichmüller space. In this paper it is proved that the Grunsky map induces a holomorphic map on the asymptotic Teichmüller space. The Caratháodory and Kobayashi metrics on the asymptotic Teichmüller space are studied as applications.