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'(0) = 1 $ that satisfy the relation $Re\,\{f'(z) + \alpha z f''(z)\} > 0$. 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 $\bigl|\,\gamma_2 \,\bigr|-\bigl|\,\gamma_1\,\bigr|$ and $\bigl|\,\Gamma_2 \,\bigr|-\bigl|\,\Gamma_1\,\bigr|$ 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 $\mathcal{W}{(1)}$.