  For a stable marginally outer trapped surface (MOTS) in an axially symmetric
spacetime with cosmological constant $\Lambda > 0$ and with matter satisfying
the dominant energy condition, we prove that the area $A$ and the angular
momentum $J$ satisfy the inequality $8\pi |J| \le A\sqrt{(1-\Lambda
A/4\pi)(1-\Lambda A/12\pi)}$ which is saturated precisely for the extreme
Kerr-deSitter family of metrics. This result entails a universal upper bound
$|J| \le J_{\max} \approx 0.17/\Lambda$ for such MOTS, which is saturated for
one particular extreme configuration. Our result sharpens the inequality $8\pi
|J| \le A$, [7,14] and we follow the overall strategy of its proof in the sense
that we estimate the area from below in terms of the energy corresponding to a
"mass functional", which is basically a suitably regularised harmonic map
$\mathbb{S}^2 \rightarrow \mathbb{H}^2 $. However, in the cosmological case
this mass functional acquires an additional potential term which itself depends
on the area. To estimate the corresponding energy in terms of the angular
momentum and the cosmological constant we use a subtle scaling argument, a
generalised "Carter-identity", and various techniques from variational
calculus, including the mountain pass theorem.
