  Using functional renormalization group we investigated possible
superconductivity in doped Sr$_2$IrO$_4$. In the electron doped case, a
$d^*_{x^2-y^2}$-wave superconducting phase is found in a narrow doping region.
The pairing is driven by spin fluctuations within the single conduction band.
In contrast, for hole doping an $s^*_{\pm}$-wave phase is established,
triggered by spin fluctuations within and across the two conduction bands. In
all cases there are comparable singlet and triplet components in the pairing
function. The Hund's rule coupling reduces (enhances) superconductivity for
electron (hole) doping. Our results imply that hole doping is more promising to
achieve a higher transition temperature. Experimental perspectives are
discussed.
