Let $\mathbb{D}$ be the unit disk on the plane and let $U,V\subset \mathbb{D}$ be open.

Is there a holomorphic map $\varphi:\mathbb{D}\times \mathbb{D}\to \mathbb{D}$ such that $U\cup V\subset \varphi(U\times V)$?

Of course one can ask the same question for "ambient domains" other than $\mathbb{D}$.