Coefficient estimates and Bohr phenomenon for analytic functions involving semigroup generator

This article investigates the Bohr phenomenon and sharp coefficient problems for the class $\mathcal{A}_{\beta}$, a subclass of analytic self-maps of the unit disk with the holomorphic generators of one-parameter continuous semigroups. By integrating concepts from complex dynamics and geometric function theory, we derive sharp improvements to the classical Bohr radius by incorporating multiple Schwarz functions and certain functional expressions. We establish generalized versions of the Bohr and Bohr-Rogosinski inequalities and determine the best possible radii for these refinements. Furthermore, we provide a sharp solution to the classical Fekete-Szeg\"o problem for the class $\mathcal{A}_{\beta}$ by obtaining sharp bounds for the functional $|a_3 - \mu a_2^2|$ for all real values of $\mu$. Additionally, we derive sharp inequalities for the moduli of differences of logarithmic coefficients for both the functions and their inverses in this class.