f = 2 Sin[Exp[z]] - Cos[Pi z] - 1/3; rts = Reduce[f == 0 && Abs[z] < 1, z]

ContourPlot[{Re[f /. z -> x + I y] == 0, Im[f /. z -> x + I y] == 0}, {x, -1, 1}, {y, -1, 1}, RegionFunction -> (Abs[#1 + I #2] < 1 &), Epilog -> {Red, PointSize[0.025], Point[{Re[z], Im[z]}] /. {ToRules[rts]}}]