Let $G\subseteq \mathbb{C}^n$ be a bounded domain. Consider the Carathéodory metric $C\_G$ on $G$. If $G=\mathbb{D}^n$ (unit polydisc), then $C\_G(a,z)=\max\_{1\leq j\leq n}p(a\_j,z\_j)$, where $p$ denotes the Poincaré metric on $\mathbb{D}$.


My question: Is there a similar formula in case $D=\mathbb{B}^2\times \mathbb{D}\subseteq \mathbb{C}^3$, where $\mathbb{B}^2$ denotes the unit ball in $\mathbb{C}^2$? More generally, is there a formula (or some estimate) for the Carathéodory metric of the product domain, in terms of the Carathéodory metric on the components?



Carathéodory metric on product domain
For a general formula, you should check out Kobayashi's book *Hyperbolic Complex Spaces*. In Proposition 3.1.11 of that text, he proves the following:



> 
> Let $X$ and $Y$ be complex spaces. For $(x,y), (x',y') \in X\times Y$, we have
> $$
> \text{max}\{ c\_X(x,x'),c\_Y(y,y')\} \leq c\_{X\times Y}((x,y),(x',y')) \leq c\_X(x,x') + c\_Y(y,y').
> $$
> 
> 
> 


