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In the present paper, we consider a subclass of starlike functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">G</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula> defined by the ratio of analytic representations of convex and starlike functions. The main aim is to determine the bounds of Fekete–Szegö-type inequalities and Hankel determinants for functions in this class. It is proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">max</mo><mfenced separators="" open="{" close="}"><mfenced separators="" open="|" close="|"><msub><mi mathvariant="script">H</mi><mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mfenced><mo>:</mo><mi>f</mi><mo>∈</mo><msub><mi mathvariant="script">G</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msub></mfenced></mrow></semantics></math></inline-formula> is equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mn>1</mn><mn>81</mn></mfrac></semantics></math></inline-formula>. The bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi mathvariant="script">G</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msub></mrow></semantics></math></inline-formula> are sharp.