A counterexample to the Clunie--Sheil-Small coefficient-difference conjecture via quasiconformal maps

We disprove the Clunie–Sheil-Small conjecture $\mid \mathrm{\mid }{a}_n\mathrm{\mid }-\mathrm{\mid }{b}_n\mathrm{\mid }\mid \le n$ for normalized univalent harmonic functions $f=h+\bar{g}$ 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 $\mathrm{\mid }{a}_n\mathrm{\mid }-\mathrm{\mid }{b}_n\mathrm{\mid }\asymp n^{1+{\epsilon }_K}$ as $n\to \mathrm{\infty }$, with ${\epsilon }_K>0$. 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 $0<p<1\mathrm{/}(2+{\epsilon }_K)$. Thus, harmonic quasiconformal maps do not retain the full conformal Hardy range $p<1\mathrm{/}2$. Interestingly, the latter range is recovered when the oscillation of the analytic dilatation on disks of fixed hyperbolic radius tends to zero near the boundary. This holds, in particular, when the dilatation has finite Dirichlet energy. For general $K$-quasiconformal maps, we impose the additional assumption that their Beltrami coefficients belong to the Sobolev space $W^{1,s}(\mathbb{D})$ for some $s\ge 1$. Every such map has bounded $p$-integral means for $0<p<1\mathrm{/}(2K)$ when $1\le s<2$, and for $0<p<1\mathrm{/}2$ when $s\ge 2$. Both these ranges are sharp. For $s\ge 2$, each map also has the same critical Hardy exponent as a conformal map onto the same image.