  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.
