Liu–et al. coefficient conjectures for univalent log-harmonic mappings

Let $f(z)=zh(z)\overline{g(z)}\in\mathcal{S}_{Lh}$, where

$$h(z)=\exp\left(\sum_{n=1}^{\infty}a_nz^n\right),\qquad g(z)=\exp\left(\sum_{n=1}^{\infty}b_nz^n\right).$$

Liu–et al.'s coefficient conjectures. For every integer $n\geq 1$, the coefficients satisfy

$$|a_n|\leq 2+\frac{1}{n},\qquad |b_n|\leq 2-\frac{1}{n},\qquad |a_n-b_n|\leq\frac{2}{n}.$$

These conjectures concern sharp coefficient bounds for the analytic and co-analytic factors of univalent log-harmonic mappings. The source attributes them to Liu et al.; their resolution is not established in the supplied text.