Liu–Ponnusamy conjectures on Bohr radii for harmonic quasi-subordination

Let $\mathbb D$ be the unit disk. Suppose that

$$f(z)=h(z)+\overline{g(z)}=\sum_{n=0}^{\infty}a_nz^n+\overline{\sum_{n=2}^{\infty}b_nz^n}$$

is a sense-preserving harmonic mapping in $\mathbb D$ and $h\prec\varphi$.

Liu–Ponnusamy's conjectures. (a) If $\varphi$ is univalent and convex in $\mathbb D$, then

$$\sum_{n=1}^{\infty}|a_n|r^n+\sum_{n=2}^{\infty}|b_n|r^n\leq \operatorname{dist}(\varphi(0),\partial\varphi(\mathbb D))$$

for $|z|=r\leq\rho_c=0.299823\cdots$, where $\rho_c$ is the positive root of

$$\frac{4r}{1-r}+2\ln(1-r)=1.$$

(b) If $\varphi$ is univalent in $\mathbb D$, then the inequality in (a) holds for $|z|=r\leq\rho_s=0.161353\cdots$, where $\rho_s$ is the positive real root of

$$\frac{2r^2}{(1-r)^2}-\ln(1-r)=\frac14.$$

These conjectures concern sharp Bohr-type radii for sense-preserving harmonic mappings, extending the corresponding finite-distortion results to the limiting case $K\to\infty$; their resolution is not established in the supplied text.