Code:

mapregion[f_, region_, xrange_, yrange_] :=
Module[{real, imag, rplot, transform, newplot},
real = ComplexExpand[Re[f /. z -> x + I*y]];
imag = ComplexExpand[Im[f /. z -> x + I*y]];
rplot = RegionPlot[region, {x, xrange[[1]], xrange[[2]]},
{y, yrange[[1]], yrange[[2]]}, PlotPoints -> 75,
AxesLabel -> {Style["x", 20], Style["y", 20]}, Frame -> None,
Axes -> True, AspectRatio -> 1, PlotRange -> {xrange, yrange}];
transform = rplot /. GraphicsComplex[pnts_, data__] :>
GraphicsComplex[({real, imag} /. {x -> #1[[1]], y -> #1[[2]]} & ) /@
pnts, data]; transform = transform /. RGBColor[x_, y_, z_] ->
RGBColor[1, 0, 0]; newplot = Show[{transform},
PlotRange -> {xrange, yrange}, AxesLabel -> {Style["u", 20],
Style["v", 20]}, Frame -> None, Axes -> True];
GraphicsArray[{{rplot, newplot}}]];
mapregion[z^3, Inequality[0, Less, Abs[x + I*y], LessEqual, 2/3] &&
0 <= Arg[x + I*y] <= Pi/4, {-1, 1}, {-1, 1}]