  Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet
problem is uniquely solvable under the curvature conditions $-C e^{2-\eta}r(x)
\leq K_M(x)\leq -1$ with $\eta>0$. We give an analytical proof of the same
statement. In addition, using this new approach we are able to establish two
boundary Harnack inequalities under the curvature condition $-C
e^{(2/3-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. This implies that there
is a natural homeomorphism between the Martin boundary and the geometric
boundary of $M$. As far as we know, this is the first result of this kind under
unbounded curvature conditions. Our proofs are modifications of arguments due
to M. T. Anderson and R. Schoen.
