The Bohr Phenomenon for Close-to-Convex Harmonic Mappings

Abstract

The classical Bohr inequality states that if $f(z)=\sum_{n=0}^{\infty} a_n z^n$ is analytic and $|f(z)|<1$ in the unit disk $\mathbb{D}$, then $\sum_{n=0}^{\infty} |a_n| r^n \le 1$ for $|z|=r \le 1/3$, where $1/3$ is sharp. Extending this to harmonic mappings $f=h+\overline{g}$ 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 $f=h+\overline{g}$ in $\mathbb{D}$ normalized by $h(0)=g(0)=h'(0)-1=g'(0)=0$. We introduce: \[ \mathcal{P}_{\mathcal{H}_0}(M) := \{ f \in \mathcal{H}_0 : \text{Re}(zh''(z)) > -M + |zg''(z)|, \; z \in \mathbb{D}, \; M > 0 \} \] \[ \mathcal{W}_{\mathcal{H}_0}(\alpha,\beta) := \{ f \in \mathcal{H}_0 : \text{Re}(h'(z) + \alpha zh''(z) - \beta) > |g'(z) + \alpha zg''(z)|, \; z \in \mathbb{D} \} \] where $\alpha \ge 0$, $\beta < 1$. We prove generalized Bohr inequalities by replacing the basis $\{r^n\}$ with non-negative continuous functions $\{\varphi_n(r)\}$. 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.