Reproduce and adapt a published result

These two examples begin with a recent coefficient claim and ask a practical question: can the package reproduce it from the class definition? One claim fails on the canonical member; the other follows from a classical geometric argument.

Run it

python examples/research_workflows/literature_coefficient_audit.py \
  --output /tmp/gfa-literature

What you will see

The JSON file compares the paper's stated value with coefficients derived from the defining subordination. The SVG shows a one-parameter family and marks where it crosses the proposed bound.

Case 1: repair a fifth-coefficient bound

Kumar and Giri study the class

\[ \frac{zf'(z)}{f(z)}\prec\frac{1+z}{\cos z}. \]

Theorem 12 of arXiv:2412.04819v1 states \(|a_5|\leq1/3\).[2] Start with the canonical member

\[ f_0(z)=z\exp\!\left( \int_0^z\frac{(1+t)/\cos t-1}{t}\,dt \right). \]

By construction,

\[ \frac{zf_0'(z)}{f_0(z)}=\frac{1+z}{\cos z}, \]

so \(f_0\) belongs to the class. Expanding directly gives

\[ f_0(z)=z+z^2+\frac34z^3+\frac7{12}z^4+\frac5{12}z^5+\cdots. \]

Thus \(a_5=5/12>1/3\). The proposed bound is false.

Find the correct bound

Write

\[ p(z)=1+\sum_{k\ge1}p_kz^k, \qquad \operatorname{Re}p(z)>0. \]

The coefficient identity used in the paper is

\[ 192a_5=-p_1^4+4p_1^2p_2+4p_1p_3+24p_4. \]

Set \(t=|p_1|\). The Carathéodory bounds and the coupled estimate

\[ \left|p_2-\frac{p_1^2}{2}\right| \leq2-\frac{t^2}{2} \]

give

\[ 192|a_5| \leq48+8t+8t^2-t^4. \]

The right-hand side increases on \(0\leq t\leq2\), so

\[ |a_5|\leq\frac5{12}. \]

The canonical member attains equality. The corrected bound is therefore sharp.

Case 2: solve an all-order conjecture geometrically

Majumder, Sarkar and Ahamed consider

\[ \frac{zf'(z)}{f(z)}\prec e^z \]

and conjecture

\[ |\gamma_n|\leq\frac1{2n} \]

for every \(n\geq1\), where \(\log(f(z)/z)=2\sum_{n\ge1}\gamma_nz^n\).[1]

The key fact is that \(e^z\) is convex and univalent on the unit disk. If \(q(z)=1+\sum b_kz^k\prec e^z\), the classical coefficient theorem for a convex dominant gives \(|b_n|\leq1\). Since

\[ q(z)=\frac{zf'(z)}{f(z)} =1+2\sum_{n\ge1}n\gamma_nz^n, \]

we immediately obtain

\[ |\gamma_n|\leq\frac1{2n}. \]

Equality is reached by functions satisfying \(zf_n'(z)/f_n(z)=e^{z^n}\). The all-order result comes from convexity, so there is no need to optimize each coefficient separately.

What to take away

For a claim from the literature, begin with the defining equation rather than a printed coefficient table. Test the canonical member first. If a claim fails, look for the geometric constraint that controls the corrected extremal problem. Before publishing a correction, check later paper versions, journal versions, and errata.

Sources

[1] https://arxiv.org/abs/2511.03218v1

[2] https://arxiv.org/abs/2412.04819v1

[3] https://matwbn.icm.edu.pl/ksiazki/apm/apm57/apm5727.pdf