The Bohr inequality for certain harmonic mappings

Let $\phi$ be analytic and univalent ({\it i.e.,} one-to-one) in $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ such that $\phi(\mathbb{D})$ has positive real part, is symmetric with respect to the real axis, starlike with respect to $\phi(0)=1,$ and $\phi ' (0)>0$. A function $f \in \mathcal{C}(\phi)$ if $1+ zf''(z)/f'(z) \prec \phi (z),$ and $f\in \mathcal{C}_{c}(\phi)$ if $2(zf'(z))'/(f(z)+\overline{f(\bar{z})})' \prec \phi (z)$ for $ z\in \mathbb{D}$. In this article, we consider the classes $\mathcal{HC}(\phi)$ and $\mathcal{HC}_{c}(\phi)$ consisting of harmonic mappings $f=h+\overline{g}$ of the form $$ h(z)=z+ \sum \limits_{n=2}^{\infty} a_{n}z^{n} \quad \mbox{and} \quad g(z)=\sum \limits_{n=2}^{\infty} b_{n}z^{n} $$ in the unit disk $\mathbb{D}$, where $h$ belongs to $\mathcal{C}(\phi)$ and $\mathcal{C}_{c}(\phi)$ respectively, with the dilation $g'(z)=\alpha z h'(z)$ and $|\alpha|<1$. Using the Bohr phenomenon for subordination classes \cite[Lemma 1]{bhowmik-2018}, we find the radius $R_{f}<1$ such that Bohr inequality $$ |z|+\sum_{n=2}^{\infty} (|a_{n}|+|b_{n}|)|z|^{n} \leq d(f(0),\partial f(\mathbb{D})) $$ holds for $|z|=r\leq R_{f}$ for the classes $\mathcal{HC}(\phi)$ and $\mathcal{HC}_{c}(\phi)$ . As a consequence of these results, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes.
Comment: 18 pages, 4 figures