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

2026-08-16

AI Summary

This study addresses computational inconsistencies and literature errors in geometric function theory by developing an atlas software system featuring directed radius graph representations and verifiable exact certificate mechanisms. Leveraging analytic control, Ma-Minda extremal constructions, and symbolic computation, the project achieves standardization and automated solving for function classes. It establishes nineteen exact sharp radii, corrects a 7.45% error in the sine-to-modified-sigmoid radius, verifies 216 Fekete-Szegő coefficient values, and compiles 702 comparative datasets. These contributions effectively unify multi-named function class systems and rectify existing constant errors, providing a rigorous computational framework for resolving longstanding discrepancies in radius and coefficient calculations within the field.

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--Szegő 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.

Problem

• Geometric Function Theory
• Inclusion Radius
• Ma-Minda Class
• Subordination
• Coefficient Problems

Innovation

• Geometric Function Atlas
• Directed Radius Problems
• Coefficient Engine
• Sharp Inclusion Radii
• Ma-Minda Generator