The Geometric Function Atlas: A Software System for Radius and Coefficient Problems

Abstract

Let $\D=\{z\in\C:|z|<1\}$, and let $\mathcal A$ be the class of analytic functions normalized by $f(0)=0$ and $f'(0)=1$. For an admissible Ma--Minda generator $\varphi$, write $\Sstar{\varphi}=\{f\in\mathcal A:zf'(z)/f(z)\prec\varphi(z)\}$. Given two generators $\varphi_1$ and $\varphi_2$, we study the largest $R\in(0,1]$ for which $f(rz)/r\in\Sstar{\varphi_2}$ whenever $f\in\Sstar{\varphi_1}$ and $0<r\le R$. We present the Geometric Function Atlas, a software system that records these directed radius problems and coefficient problems by their exact generators, parameter domains, normalizations, and sharpness statements. This representation identifies the same class across alternative names and transliterations while keeping the two directions of an inclusion problem distinct. The coefficient engine recovers all 216 Fekete--Szeg\H{o} values predicted by the general Ma--Minda formula across 36 registered classes. The directed-radius atlas contains 702 ordered comparisons; omitting direction merges unequal constants in 253 of the 262 class-pair families represented in both directions. Using boundary contact, analytic majorants, and explicit Ma--Minda extremals, we prove nineteen exact sharp inclusion radii. In particular, the sine-to-modified-sigmoid radius is $\arcsin((e-1)/(e+1))$, improving the published sufficient radius $\operatorname{arsinh}((e-1)/(e+1))$ by 7.45\%. For the crescent and exponential classes, the reciprocal sharp radii are $\sin1$ and $\log(1+\sqrt2)$; the latter corrects a published constant. The Python package, exact certificates, and registry records accompany the paper.