The classical Bohr inequality states that if $ f $ is an analytic function with the power series representation $ f(z)=\sum_{ n=0}^{ \infty}a_nz^n $ in the unit disk $ \mathbb{ D}:=\{ z\in\mathbb{ C} : |z|<1\} $ such that $ |f(z)|\leq 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\leq 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}^{ 0}_{ \mathcal{ H}}(M) :=\{ f=h+\overline{ g} \in \mathcal{ H}_{ 0}: \real (zh^{ \prime\prime}(z))> -M+|zg^{ \prime\prime}(z)|,\; z \in \mathbb{ D},\; M>0\} $. Functions in the class $ \mathcal{ P}^{ 0}_{ \mathcal{ H}}(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) $.
Categories: math.CV