  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 the
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 of 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, a 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.
