Geometric Studies and the Bohr Radius for Certain Normalized Harmonic Mappings

Let \(\mathcal {H}\) $\mathcal{H}$ be the class of harmonic functions \(f=h+\overline{g}\) $f=h+\overline{g}$ in the unit disk \(\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}\) $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ , where h and g are analytic in \(\mathbb {D}\) $\mathbb{D}$ . In 2020, N. Ghosh and V. Allu introduced the class \(\mathcal {P}_{\mathcal {H}}^0(M)\) ${\mathcal{P}}_{\mathcal{H}}^0(M)$ of normalized harmonic mappings defined by \(\mathcal {P}_{\mathcal {H}}^0(M)=\{f=h+\overline{g}\in \mathcal {H}: \text {Re}(zh''(z))>-M+|zg''(z)|\;\text {with}\;M>0, g'(0)=0, z\in \mathbb {D}\}\) ${\mathcal{P}}_{\mathcal{H}}^0(M)=\{f=h+\overline{g}\in \mathcal{H}:\text{Re}(z{h}^{\prime \prime }(z))>-M+|z{g}^{\prime \prime }(z)|\text{with}M>0,g^{\prime }(0)=0,z\in \mathbb{D}\}$ . In this paper, we investigate various geometric properties such as starlikeness, convexity, convex combination and convolution for functions in the class \(\mathcal {P}_{\mathcal {H}}^0(M)\) ${\mathcal{P}}_{\mathcal{H}}^0(M)$ . Furthermore, we determine the sharp Bohr–Rogosinski radius, improved Bohr radius and refined Bohr radius for the class \(\mathcal {P}_{\mathcal {H}}^0(M)\) ${\mathcal{P}}_{\mathcal{H}}^0(M)$ .