| Summary: | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula> defined in the new domain, including the sharp estimates for the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>4</mn></msub></semantics></math></inline-formula>, and for three second-order and third-order Hankel determinants, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">H</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub><mi mathvariant="script">X</mi><mo>,</mo><msub><mi mathvariant="script">H</mi><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mi mathvariant="script">X</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">H</mi><mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msub><mi mathvariant="script">X</mi></mrow></semantics></math></inline-formula>. The optimality of each obtained estimate is given as well.
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