Improved Bohr inequalities for certain class of harmonic univalent functions

Abstract

Let $\mathcal{H}$ be the class of complex-valued harmonic mappings $f=h+\bar{g}$ defined in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic functions in $\mathbb{D}$ with the normalization $h(0)=0=h^{\prime }(0)-1$ and $g(0)=0$. Let ${\mathcal{H}}_0=\{f=h+\bar{g}\in \mathcal{H}:g^{\prime }(0)=0\}.$ Ghosh and Vasudevrao \cite{Ghosh-Vasudevarao-BAMS-2020} have studied the following interesting harmonic univalent class ${\mathcal{P}}_{\mathcal{H}}^0(M)$ which is defined by $$\mathcal{P}^{0}_{\mathcal{H}}(M) :=\{f=h+\overline{g} \in \mathcal{H}_{0}: \mathrm{Re} (zh^{\prime\prime}(z))> -M+|zg^{\prime\prime}(z)|,\; z \in \mathbb{D}\; \mbox{and}\;\; M>0\}. $$ In this paper, we obtain the sharp Bohr-Rogosinski inequality, improved Bohr inequality, refined Bohr inequality and Bohr-type inequality for the class ${\mathcal{P}}_{\mathcal{H}}^0(M)$.

19 pages, 3 figures. arXiv admin note: text overlap with arXiv:2012.06829