Mass of Dyonic Black Holes and Entropy Super-Additivity

We study extremal static dyonic black holes in four-dimensional Einstein-Maxwell-Dilaton theory, for general values of the constant $a$ in the exponential coupling $e^{a\phi}$ of the dilaton to the Maxwell kinetic term. Explicit solutions are known only for $a=0$, $a=1$ and $a=\sqrt3$, and for general $a$ when the electric and magnetic charges $Q$ and $P$ are equal. We obtain solutions as power series expansions around $Q=P$, in terms of a small parameter $\epsilon= a^{-1}\, \log(Q/P)$. Using these, and also solutions constructed numerically, we test a relation between the mass and the charges that had been conjectured long ago by Rasheed. We find that although the conjecture is not exactly correct it is in fact quite accurate for a wide range of the black hole parameters. We investigate some improved conjectures for the mass relation. We also study the circumstances under which entropy super-additivity, which is related to Hawking's area theorem, is violated. This extends beyond previous examples exhibited in the literature for the particular case of $a=\sqrt3$ dyonic black holes.