Example

Find the image of the unit disk |z|<1under the tramsformation \(TraditionalForm\`z\  -> \ z\^2\  - \ z\&_\  + \ 1\).
Show that \(TraditionalForm\`\(\(\ \)\(\(|\)\(z\^2\  - \ z\&_\  + \ 
          1\)\(|\)\(\(<\)\(3\)\)\)\)\) if |z|<1.

x  = r  Cos[t];  y  =  r Sin[t];  
    z  =  x  +  I y;
w = z^2  -  Conjugate[z]  +  1;
Do[p[r] = 
        ParametricPlot[{Re[w], Im[w]}, {t, 0, 2  π},  
          AspectRatio -> Automatic, 
          PlotRange -> {{-1, 3}, {-2, 2}}], {r, 
        1,  .1, - .1}]

Show[Table[p[r], {r, 1,  .1, - .1}]]

Of course, the triangle inequality shows: \(TraditionalForm\`\(\(|\)\(z\^2\  - \ z\&_\  + \ 
          1\)\(|\)\(\ \)\(\(≤\)\(\ \)\(\(|\)\(z\)\( | \^2\)\(\ \)\(+\  | \)\)\)\)\)\(TraditionalForm\`z\&_ | \ \(+\ 1\)\  < \ 1\^2 + 1 + 1 = 3\)

Example

Find the image of z  ->  (1/((z^4) - 4  (z^2) + 3)) for |z|  =  2. Show |(1/((z^4) - 4  (z^2) + 
              3))|<(1/3).

x  = r  Cos[t];  y  =  r Sin[t];  
    z  =  x  +  I y;
w = 1/(z^4  - 4  z^2  +  3) /. r -> 2;
ParametricPlot[{Re[w], Im[w]}, {t, 0, π},  
      AspectRatio -> Automatic]

Note that (z^4) - 4  (z^2) + 
          3  =  ((z^2) - 3) ((z^2) - 1) and
 \(TraditionalForm\`\(\(|\)\(z\^2 - 
            3\)\(|\)\(\ \)\(\(≥\)\(\(||\)\(z\)\)\)\( | \^2\)\(-\ 3\)\(|\)\)\  = \ 1\) while
 \(TraditionalForm\`\(\(|\)\(z\^2 - 
            1\)\(|\)\(\ \)\(\(≥\)\(\(||\)\(z\)\)\)\( | \^2\)\(-\ 1\)\(|\)\)\  = \ 3\) since \(TraditionalForm\`\(\(\(|\)\(z\)\(|\)\)\(\ \)\(=\)\(\ \)\(2\)\(\ \)\)\). Therefore
 \(TraditionalForm\`\(\(|\)\(1\/\(z\^4 - 4  z\^2 + 
                3\)\)\(|\)\) = \ \(1\/\(\(|\)\(z\^2 - 3\)\(|\)\(\ \)\(|\)\(z\^2 - 1\)\(|\)\) ≤ \(1\/1\) 1\/3\  = \ 1\/3\)\).