Logarithmic Coefficients Problems of Geometric Subclass of Closed-to-convex Functions

For $\alpha \ge 0$, let $\mathcal{W}(\alpha )$ be the class of all analytic functions in the unit disk $\mathbb{D}$ with normalization $f(0)=0$ and $f^{\mathrm{\prime }}(0)=1$ that satisfy the relation $Re\{f^{\mathrm{\prime }}(z)+\alpha z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\}>0$. This article aims to establish sharp bounds for logarithmic coefficients ${\gamma }_1$, ${\gamma }_2$ and ${\gamma }_3$ and logarithmic inverse coefficients ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$ and ${\mathrm{\Gamma }}_3$ of functions in $\mathcal{W}(\alpha )$. The sharp upper and lower bounds for $\mid {\gamma }_2\mid -\mid {\gamma }_1\mid$ and $\mid {\mathrm{\Gamma }}_2\mid -\mid {\mathrm{\Gamma }}_1\mid$ have been obtained for the class $\mathcal{W}(\alpha )$. In addition, we establish sharp inequality for the second Hankel determinant of the logarithmic and inverse logarithmic coefficients for the class $\mathcal{W}(1)$.