  We prove area inequalities for stable marginally outer trapped surfaces in
Einstein-Maxwell-dilaton theory. Our inspiration comes on the one hand from a
corresponding upper bound for the area in terms of the charges obtained
recently by Dain, Jaramillo and Reiris [1] in the pure Einstein-Maxwell case
without symmetries, and on the other hand from Yazadjiev's inequality [2] in
the axially symmetric Einstein-Maxwell-dilaton case. The common issue in these
proofs and in the present one is a functional ${\mathscr W}$ of the matter
fields for which the stability condition readily yields an {\it upper} bound.
On the other hand, the step which crucially depends on whether or not a dilaton
field is present is to obtain a {\it lower} bound for ${\mathscr W}$ as well.
We obtain the latter by first setting up a variational principle for ${\mathscr
W}$ with respect to the dilaton field $\phi$, then by proving existence of a
minimizer $\psi$ as solution of the corresponding Euler-Lagrange equations and
finally by estimating ${\mathscr W}(\psi)$. In the special case that the normal
components of the electric and magnetic fields are proportional we obtain the
area bound $A \ge 8\pi P Q$ in terms of the electric and magnetic charges. In
the generic case our results are less explicit but imply rigorous
`perturbation' results for the above inequality. All our inequalities are
saturated for a 2-parameter family of static, extreme solutions found by
Gibbons [3]. Via the Bekenstein-Hawking relation $A = 4S$ our results give
positive lower bounds for the entropy $S$ which are particularly interesting in
the Einstein-Maxwell-dilaton case.
