Geometric Studies and the Bohr Radius for Certain Normalized Harmonic Mappings

Abstract

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