Das–Kaliraj coefficient-growth conjecture for univalent harmonic mappings

Let $\mathcal{S}_{H}$ be the family of sense-preserving planar harmonic univalent mappings $f=h+\overline{g}$ in $\mathbb{D}$ satisfying $h(0)=g(0)=h'(0)-1=0$, and let

$$\gamma=\sup_{f\in\mathcal{S}_{H}}\frac{|f_{zz}(0)|}{2}.$$

Assume the hypotheses of Theorem E: $f=h+\overline{g}\in\mathcal{S}_{H}$ with the normalized expansions used there. Das–Kaliraj's coefficient-growth conjecture. There is an absolute constant $C$ such that

$$|a_n|\leq Cn^{\gamma-1}\qquad\text{for }n\in\{2,3,\ldots\}.$$

The conjecture seeks the coefficient-growth order suggested by the harmonic Bieberbach conjecture and improves the previously stated $n^{\gamma-1/2}$ estimate. Its status is not resolved in the supplied text.