Let $U\subset\mathbb{C}^n$ be a domain of holomorphy.
The following proposition is true?
For each $a\in\partial U$, there is a holomorphic function $f\in H(U)$, such that $\displaystyle\sup_{j\ge 1} |f(z_j)|=\infty$ $($for all sequence $ (z_j)_{j\ge 1}\subset U$ such that $\lim_{j\to\infty} z_j=a)$.
Any hint would be appreciated.
The basic idea is of course: $f(z) = \frac{1}{\sum_{i = 1}^n (z_i - a_i)}$ and the point is that this works with only a minor twist for reasonably nice domains. I would start by considering $U$ that are convex and have smooth boundary and try to find a particular real valued function on this domain that has some nice properties on this domain, then use some constants one can obtain from this function to balance out the summands here such that this function does not blow up on $U$.