Alexander transform

The Alexander transform is a useful identity connecting starlike and convex functions. It is kept here as an advanced example rather than presented as a separate research workflow.

The identity

For a normalized analytic function \(f\), define

\[ g(z)=\int_0^z\frac{f(t)}{t}\,dt. \]

Then \(zg'(z)=f(z)\) and

\[ 1+\frac{zg''(z)}{g'(z)}=\frac{zf'(z)}{f(z)}. \]

The right-hand side tests starlikeness of \(f\); the left-hand side tests convexity of \(g\). If

\[ f(z)=z+\sum_{n\ge2}a_nz^n, \]

then

\[ g(z)=z+\sum_{n\ge2}\frac{a_n}{n}z^n. \]

Worked example

From a source checkout, run:

python examples/research_workflows/alexander_transform.py \
  --generator starlike \
  --output /tmp/gfa-alexander

The output contains the exact transformed coefficients, a symbolic check of the identity, and a sampled before-and-after plot. For the Koebe function \(f(z)=z/(1-z)^2\), the transform is \(g(z)=z/(1-z)\).