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

Abstract

For $α\ge 0$, let $\mathcal{W}(α)$ 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 $Re\,\{f'(z) + αz f''(z)\} > 0$. This article aims to establish sharp bounds for logarithmic coefficients $γ_1$, $γ_2$ and $γ_3$ and logarithmic inverse coefficients $Γ_1$, $Γ_2$ and $Γ_3$ of functions in $\mathcal{W}(α)$. The sharp upper and lower bounds for $\bigl|\,γ_2 \,\bigr|-\bigl|\,γ_1\,\bigr|$ and $\bigl|\,Γ_2 \,\bigr|-\bigl|\,Γ_1\,\bigr|$ have been obtained for the class $\mathcal{W}{(α)}$. In addition, we establish sharp inequality for the second Hankel determinant of the logarithmic and inverse logarithmic coefficients for the class $\mathcal{W}{(1)}$.