The sharp refined Bohr inequalities for a subclass of close-to-convex harmonic mappings

Let $\mathcal{H}$ be the class of normalized complex valued harmonic functions $f=h+\bar{g}$ defined on the unit disk $\mathbb{D}$, where $h$ and $g$ are analytic functions with the normalization conditions $h(0)=h^{\mathrm{\prime }}(0)-1=0$ and $g(0)=0$. For the class $R_H^0(\gamma ,\delta ,\lambda )$ ( $0\le \lambda <\gamma \le \delta$) consisting of functions $f=h+\bar{g}\in \mathcal{H}$ satisfying the condition $f_{\bar{z}}(0)=0$ and the inequality $Re(\gamma h^{\mathrm{\prime }}(z)+\delta z{h}^{\mathrm{\prime }\mathrm{\prime }}(z)+(\frac{\delta -\gamma }{2})z^2{h}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(z)-\lambda )>\mathrm{\mid }\gamma g^{\mathrm{\prime }}(z)+\delta z{g}^{\mathrm{\prime }\mathrm{\prime }}(z)+(\frac{\delta -\gamma }{2})z^2{g}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{\mid }$, we obtain sharp improved Bohr Phenomenon, refined Bohr radius and the Bohr-Rogosinski inequality for the class $R_H^0(\gamma ,\delta ,\lambda )$.