​Analytic Functions Related to a Balloon-Shaped Domain. Fractal Fract. 2023, 7, 865. https:// Abstract: One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X defined in the new domain, including the sharp estimates for the coefficients a 2 , a 3 , and a 4 , and for three second-order and third-order Hankel determinants, H 2,1 X , H 2,2 X , and H 3,1 X. The optimality of each obtained estimate is given as well.​