The Bohr Phenomenon for Close-to-Convex Harmonic Mappings

The classical Bohr inequality states that if $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ is analytic and $\mathrm{\mid }f(z)\mathrm{\mid }<1$ in the unit disk $\mathbb{D}$, then ${\sum }_{n=0}^{\mathrm{\infty }}\mathrm{\mid }{a}_n\mathrm{\mid }{r}^n\le 1$ for $\mathrm{\mid }z\mathrm{\mid }=r\le 1\mathrm{/}3$, where $1\mathrm{/}3$ is sharp. Extending this to harmonic mappings $f=h+\bar{g}$ is central in geometric function theory due to the co-analytic part $g$. This paper establishes sharp Bohr-type inequalities for two classes of sense-preserving close-to-convex harmonic mappings. Let ${\mathcal{H}}_0$ be the class of harmonic mappings $f=h+\bar{g}$ in $\mathbb{D}$ normalized by $h(0)=g(0)=h^{\mathrm{\prime }}(0)-1=g^{\mathrm{\prime }}(0)=0$. We introduce: $${\mathcal{P}}_{{\mathcal{H}}_0}(M):=\{f\in {\mathcal{H}}_0:\text{Re}(z{h}^{\mathrm{\prime }\mathrm{\prime }}(z))>-M+\mathrm{\mid }z{g}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{\mid },z\in \mathbb{D},M>0\}$$ $${\mathcal{W}}_{{\mathcal{H}}_0}(\alpha ,\beta ):=\{f\in {\mathcal{H}}_0:\text{Re}(h^{\mathrm{\prime }}(z)+\alpha z{h}^{\mathrm{\prime }\mathrm{\prime }}(z)-\beta )>\mathrm{\mid }{g}^{\mathrm{\prime }}(z)+\alpha z{g}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{\mid },z\in \mathbb{D}\}$$ where $\alpha \ge 0$, $\beta <1$. We prove generalized Bohr inequalities by replacing the basis $\{r^n\}$ with non-negative continuous functions $\{{\phi }_n(r)\}$. The results are proved using sharp coefficient bounds and growth theorems, providing new insights into the Bohr phenomenon for harmonic mappings and subclasses defined by differential inequalities.