Investigate a coefficient problem

For

\[ f(z)=z+a_2z^2+a_3z^3+\cdots, \]

the Fekete–Szegő problem asks for the largest possible value of \(|a_3-\mu a_2^2|\). This workflow shows how the answer changes as the real parameter \(\mu\) moves.

Run it

python examples/research_workflows/coefficient_comparison.py \
  --generator starlike \
  --output /tmp/gfa-coefficients

What you will see

Open /tmp/gfa-coefficients/coefficient_comparison.svg. The graph has a flat middle section and two sloping sides. The JSON file gives the exact transition points and an equality example for each part.

For the classical starlike class, the answer is

\[ \max\{1,|3-4\mu|\}. \]

The graph changes at \(\mu=1/2\) and \(\mu=1\).

Why the graph has corners

Write

\[ \phi(z)=1+B_1z+B_2z^2+\cdots \]

and

\[ \frac{zf'(z)}{f(z)}=\phi(\omega(z)), \qquad \omega(z)=c_1z+c_2z^2+\cdots. \]

Comparing coefficients gives

\[ a_3-\mu a_2^2 =\frac{B_1}{2}\left(c_2+Qc_1^2\right), \qquad Q=\frac{B_2}{B_1}+(1-2\mu)B_1. \]

The Schwarz coefficient inequality yields

\[ |a_3-\mu a_2^2| \leq \frac{B_1}{2}\max\{1,|Q|\}. \]

Two functions compete for equality: \(\omega(z)=z\) supplies the sloping pieces, while \(\omega(z)=z^2\) supplies the flat piece. A corner appears when the winner changes.

Try the sine class

python examples/research_workflows/coefficient_comparison.py \
  --generator sine \
  --output /tmp/gfa-sine-coefficients

Here \(B_1=1\) and \(B_2=0\), so the bound is

\[ \frac12\max\{1,|1-2\mu|\}, \]

with transitions at \(\mu=0\) and \(\mu=1\).

The formulas above prove the bounds for every real \(\mu\). The SVG simply makes the piecewise formula easier to see.

Source: W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis, Tianjin 1992, International Press (1994), 157–169.