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We establish the conjectured area-angular momentum-charge inequality forstable apparent horizons in the presence of a positive cosmological constant,and show that it is saturated precisely for extreme Kerr-Newman-de Sitterhorizons. As with previous inequalities of this type, the proof is reduced tominimizing an `area functional' related to a harmonic map energy; in this casemaps are from the 2-sphere to the complex hyperbolic plane. The proof here issimplified compared to previous results for less embellished inequalities, dueto the observation that the functional is convex along geodesic deformations inthe target.

The Area-Angular Momentum-Charge Inequality for Black Holes With Positive Cosmological Constant