Estimating 2,3-Fold Hankel Determinants, Zalcman Functionals and Logarithmic Coefficients of Certain Subclasses of Holomorphic Functions with Bounded Rotations

The study explores analytic, geometric and algebaraic properties of two subclasses of analytic functions: the class of Bounded Radius Rotation denoted by ${\mathcal{R}}_{\mathfrak{s}}{,}_{\varrho }(\mathcal{A},\mathcal{B},z)$, and the class of Bounded Boundary Rotation denoted by ${\mathcal{V}}_{\mathfrak{s}}{,}_{\varrho }(\mathcal{A},\mathcal{B},z)$, both associated with strongly Janowski type functions. In particular, we obtain upper bounds for the third-order Hankel determinant $|{\mathcal{H}}_{3,1}f\left(z\right)|$ and concentrate on functions displaying 2- and 3-fold symmetry. We also provide estimates for the initial logarithmic coefficients ${\eta }_1,{\eta }_2,{\eta }_3$ and the Zalcman functional $|t_3^2-t_5|$ for each class. These findings provide fresh insights into the behavior of generalized subclasses of univalent function.