f=(1+z)/(1-z);Manipulate[ParametricPlot[With[{z=r Exp[I t]},Through[{Re,Im}[s ff + (1-s)z]]],{r,0,1},{t,0,2Pi},Mesh->15,PlotRange->2,MeshStyle->{Red, Blue},PlotStyle->None,PlotPoints->50],{s,0,1}]/. ff->f

n=3;
F[z_?NumericQ]:=ReIm[NIntegrate[Power[(1.-(t z)^n),-2/n] z,{t,0,1},AccuracyGoal->4]]
ParametricPlot[F[r Exp[I s]],{s,-Pi,Pi},{r,0.,1.},Mesh->{6 n-1,20},PlotPoints->{6 n-1,20},MaxRecursion->1,PlotRange->2]