A counterexample to the Clunie--Sheil-Small coefficient-difference conjecture via quasiconformal maps
Year: 2026
Type: preprint
Abstract: We disprove the Clunie--Sheil-Small conjecture $\bigl||a_n|-|b_n|\bigr|\le n$ for normalized univalent harmonic functions $f=h+\overline g$ in the unit disk $\mathbb{D}$, where $a_n$ and $b_n$ are the Taylor coefficients of $h$ and $g$, respectively. For every $K>1$, we construct a harmonic $K$-quasiconformal map which has $|a_n|-|b_n|\asymp n^{1+\varepsilon_K}$ as $n\to\infty$, with $\varepsilon_K>0$. Nevertheless, this map satisfies both individual coefficient bounds in the conjecture. Next, we show that the same map belongs to the harmonic Hardy space $h^p$ if and only if $0
Source: arXiv (Cornell University)
Author: Das, Suman
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221033935