  We study the correlated electronic structure of single-layer iridates based
on structurally-undistorted Ba$_2$IrO$_4$. Starting from the first-principles
band structure, the interplay between local Coulomb interactions and spin-orbit
coupling is investigated by means of rotational-invariant slave-boson
mean-field theory. The evolution from a three-band description towards an
anisotropic one-band ($J=1/2$) picture is traced. Single-site and cluster
self-energies are used to shed light on competing Slater- and Mott-dominated
correlation regimes. We reveal a clear asymmetry between electron and hole
doping, notably in the nodal/anti-nodal Fermi-surface dichotomy at strong
coupling. Electron-doped iridates appear comparable to hole-doped cuprates due
to the different sign of the next-nearest-neighbor hopping $t'$.
