This paper presents coefficient inequalities for functions associated with hyperbolic domains. The study focuses on the Fekete-Szegö functional, which is a key tool in the analysis of univalent functions. The paper aims to determine upper bounds for this functional for certain analytic functions that map into hyperbolic regions. These bounds refine existing results and provide new insights into the behavior of such functions. Additionally, the paper extends its analysis to the inverse functions of these analytic functions, ensuring a comprehensive understanding of their properties. The results are derived using established techniques in complex analysis and are supported by detailed mathematical derivations and references to related literature. The findings contribute to the broader field of geometric function theory by enhancing the understanding of coefficient inequalities in the context of hyperbolic domains.This paper presents coefficient inequalities for functions associated with hyperbolic domains. The study focuses on the Fekete-Szegö functional, which is a key tool in the analysis of univalent functions. The paper aims to determine upper bounds for this functional for certain analytic functions that map into hyperbolic regions. These bounds refine existing results and provide new insights into the behavior of such functions. Additionally, the paper extends its analysis to the inverse functions of these analytic functions, ensuring a comprehensive understanding of their properties. The results are derived using established techniques in complex analysis and are supported by detailed mathematical derivations and references to related literature. The findings contribute to the broader field of geometric function theory by enhancing the understanding of coefficient inequalities in the context of hyperbolic domains.