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

Abstract

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 $f^{\prime }(0)=1$ that satisfy the relation $Re\{f^{\prime }(z)+\alpha z{f}^{\prime \prime }(z)\}>0$. This article aims to establish sharp bounds for logarithmic coefficients ${\gamma }_1$, ${\gamma }_2$ and ${\gamma }_3$ and logarithmic inverse coefficients ${\Gamma }_1$, ${\Gamma }_2$ and ${\Gamma }_3$ of functions in $\mathcal{W}(\alpha )$. The sharp upper and lower bounds for $\left|{\gamma }_2\right|-\left|{\gamma }_1\right|$ and $\left|{\Gamma }_2\right|-\left|{\Gamma }_1\right|$ 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 $\mathcal{W}(1)$.