  The recent holographic deduction of Penrose inequality only assumes null
energy condition while the weak or dominant energy condition is required in
usual geometric proof. This paper makes a step toward filling up gap between
these two approaches. For planar/spherically symmetrically asymptotically
Schwarzschild anti-de Sitter (AdS) black holes, we give a purely geometric
proof for Penrose inequality by assuming the null energy condition. We also
point out that two naive generalizations of charged Penrose inequality are not
generally true and propose two new candidates. When the spacetime is
asymptotically AdS but not Schwarzschild-AdS, the total mass is defined
according to holographic renormalization and depends on scheme of quantization.
In this case, the holographic argument implies that the Penrose inequality
should still be valid but this paper use concrete example to show that whether
the Penrose inequality holds or not will depend on what kind of quantization
scheme we employ.
