Bohr-Rogosinski and improved Bohr type inequalities for certain fully starlike harmonic mappings

Abstract

The classical Bohr inequality states that if $f$ is an analytic function with the power series representation $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ such that $|f(z)|\le 1$ for all $z\in \mathbb{D}$, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\; \text{for}\;\; |z|=r\leq\frac{1}{3} \end{equation*} and the constant $1/3$ cannot be improved. The constant $r_0=1/3$ is known as Bohr radius and the inequality ${\sum }_{n=0}^{\infty }|a_n|r^n\le 1$ is known as Bohr inequality. Let $\mathcal{H}$ be the class of complex-valued harmonic mappings $f=h+\bar{g}$ defined in the unit disk $\mathbb{D}$, 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 ${\mathcal{H}}_0=\{f=h+\bar{g}\in \mathcal{H}:g^{\prime }(0)=0\}.$ Let ${\mathcal{P}}_{\mathcal{H}}^0(M):=\{f=h+\overline{g}\in {\mathcal{H}}_0:\Re (z{h}^{\prime \prime }(z))>-M+|z{g}^{\prime \prime }(z)|,z\in \mathbb{D},M>0\}$. Functions in the class ${\mathcal{P}}_{\mathcal{H}}^0(M)$ are called fully starlike univalent functions for $0<M<1/\log 4$. In this paper, we obtain the sharp Bohr-Rogosinski type inequality and improved Bohr inequality and the corresponding Bohr radius for the class ${\mathcal{P}}_{\mathcal{H}}^0(M)$.

19 pages, 4 figures. arXiv admin note: text overlap with arXiv:2104.02099

References in corpus (2)

• Refined Bohr inequality for bounded analytic functions
• Improved Bohr radius for the class of starlike log-harmonic mappings