The Hankel determinant $H_{ 2,2}(F_{ f}/2)$ is defined as: \begin{ align*} H_{ 2,2}(F_{ f}/2):= \begin{ vmatrix} \gamma_2 & \gamma_3 \gamma_3 & \gamma_4 \end{ vmatrix}, \end{ align*} where $\gamma_2, \gamma_3,$ and $\gamma_4$ are the second, third, and fourth logarithmic coefficients of functions belonging to the class $\mathcal{ S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{ 2,2}(F_{ f}/2)|\leq (1272 + 113\sqrt{ 678})/32856$ and $|H_{ 2,2}(F_{ f}/2)| \leq 13/1080$ for the logarithmic coefficients of starlike and convex functions with respect to symmetric points. Moreover, we provide examples that demonstrate the strict inequality holds.
Categories: math.CV