Holomorphic maps with large images

Abstract: We show that each pseudoconvex domain $\Omega\subset {\mathbb C}^n$ admits a holomorphic map $F$ to ${\mathbb C}^m$ with $|F|\le C_1 e^{C_2 \hat{\delta}^{-6}}$, where $\hat{\delta}$ is the minimum of the boundary distance and $(1+|z|^2)^{-1/2}$, such that every boundary point is a Casorati-Weierstrass point of $F$. Based on this fact, we introduce a new anti-hyperbolic concept --- universal dominability. We also show that for each $\alpha>6$ and each pseudoconvex domain $\Omega\subset {\mathbb C}^n$, there is a holomorphic function $f$ on ${\Omega}$ with $|f|\le C_\alpha e^{C_\alpha' \hat{\delta}^{-\alpha}}$, such that every boundary point is a Picard point of $F$. Applications to the construction of holomorphic maps of a given domain onto some ${\mathbb C}^m$ are given.