  We define an attractive gravity probe surface (AGPS) as a compact 2-surface
$S_\alpha$ with positive mean curvature $k$ satisfying $r^a D_a k / k^2 \ge
\alpha$ (for a constant $\alpha>-1/2$) in the local inverse mean curvature
flow, where $r^a D_a k$ is the derivative of $k$ in the outward unit normal
direction. For asymptotically flat spaces, any AGPS is proved to satisfy the
areal inequality $A_\alpha \le 4\pi [ ( 3+4\alpha)/(1+2\alpha) ]^2(Gm)^2$,
where $A_{\alpha}$ is the area of $S_\alpha$ and $m$ is Arnowitt-Deser-Misner
(ADM) mass. Equality is realized when the space is isometric to the
$t=$constant hypersurface of the Schwarzschild spacetime and $S_\alpha$ is an
$r=\mathrm{constant}$ surface with $r^a D_a k / k^2 = \alpha$. We adapt the two
methods, the inverse mean curvature flow and the conformal flow. Therefore, our
result is applicable to the case where $S_\alpha$ has multiple components. For
anti-de Sitter (AdS) spaces, the similar inequality is derived, but the proof
is performed only by using the inverse mean curvature flow. We also discuss the
cases with asymptotically locally AdS spaces.
