= Boundary-adapted holomorphic zero factor
{title2=$E_{a,w,K}(z)$}
For a proper plane <domain> $D$, choose $a\in D$ and $w\notin D$ attaining $\delta(a)=\operatorname{dist}(a,\mathbb C\setminus D)$. The <holomorphic function>
$$
E_{a,w,K}(z)=\frac{z-a}{z-w}\exp\left(\sum_{k=1}^K\frac1k\left(\frac{a-w}{z-w}\right)^k\right)
$$
has exactly one zero in $D$, a simple zero at $a$. For $|z-w|>2\delta(a)$ its zero-free <holomorphic logarithm> satisfies
$$
\left|\log E_{a,w,K}(z)\right|\leq\sum_{k>K}\frac{2^{-k}}k\leq\frac{2^{-K}}{K+1}.
$$
Thus products of such factors can realize prescribed zeros approaching the boundary. If the zero has multiplicity $m$, choose the logarithmic error smaller by a factor $m$ before taking the $m$th power. On a compact subset of $D$, eventual uniform summability of these logarithms gives a nonvanishing product tail and a <holomorphic function> with precisely the requested zeros.
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