= Solution
First construct a <holomorphic function> without requiring boundedness. Separate the finitely many zeros at zero into $z^m$, and enumerate the remaining prescribed zeros as $a_n$. Since $|a_n|\to1$, there is no interior accumulation point and every prescribed point has finite <multiplicity>. Using the <Weierstrass elementary factors>, set
$$
F(z)=z^m\prod_{n\ge1}E_n(z/a_n),\qquad
E_n(t)=(1-t)\exp\left(\sum_{k=1}^n\frac{t^k}{k}\right).
$$
For a compact disc $|z|\le r<1$, choose $q$ with $r<q<1$. Eventually $|z/a_n|\le q$. The <power series> identity
$$
\log E_n(t)=-\sum_{k=n+1}^\infty\frac{t^k}{k},\qquad
|\log E_n(t)|\le\frac{q^{n+1}}{(n+1)(1-q)}
$$
gives an absolutely summable uniform bound. The <infinite product convergence from logarithmic tails> therefore makes the tail <holomorphic> and nonvanishing on this compact set. The finitely many initial factors have exactly their prescribed zeros, with the prescribed <multiplicities>. Hence $F$ has exactly the required zero sequence. This is a direct <canonical product construction for zeros escaping to the disk boundary>.
For a nonzero bounded <holomorphic function>, <Jensen's formula> bounds $\sum_n\log(1/|a_n|)$ after separating the zero at zero. Thus the <Blaschke condition> is necessary; the <Blaschke product> construction makes it sufficient. Use the curvature-minus-one convention for <Hyperbolic distance in the Poincare disc>:
$$
\rho(0,z)=\log\frac{1+|z|}{1-|z|},\qquad
e^{-\rho(0,z)}=\frac{1-|z|}{1+|z|}.
$$
The latter lies between $(1-|z|)/2$ and $1-|z|$. Consequently the precise bounded-function criterion is
$$
\boxed{\sum_n e^{-\rho(0,z_n)}<\infty.}
$$
This counts <multiplicities>; the finitely many occurrences at zero cause no problem.
For the requested <hyperbolic zero estimate for a Blaschke product>, the <pseudohyperbolic distance> identity is
$$
|b_a(w)|=\left|\frac{w-a}{1-\overline a w}\right|
=\tanh\frac{\rho(w,a)}2=\frac{1-t}{1+t},\qquad t=e^{-\rho(w,a)}.
$$
If $w$ is not a zero, $0<t<1$. The real function $L(t)=\log[(1+t)/(1-t)]-2t$ has $L(0)=0$ and $L'(t)=2t^2/(1-t^2)\ge0$, so $L(t)\ge0$. Thus $\log|b_a(w)|\le-2e^{-\rho(w,a)}$. Apply this to each factor, including factors at zero, and then pass from finite products to the <Blaschke product> by <locally uniform convergence>:
$$
\boxed{|B(w)|\le\exp\left(-2\sum_n e^{-\rho(w,z_n)}\right).}
$$
At a zero of $B$ the left side is zero, so the inequality is immediate; the logarithmic calculation at $t=1$ is unnecessary. The unimodular constant contributes nothing. Away from the zeros the product is nonzero, so the same inequality also bounds the sum and shows it is finite there.
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