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In our present investigation, a subclass of starlike function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi mathvariant="script">L</mi></mrow><mo>*</mo></msubsup></semantics></math></inline-formula> connected with a domain bounded by an epicycloid with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> cusps was considered. The main work is to investigate some coefficient inequalities, and second and third Hankel determinants for functions belonging to this class. In particular, we calculate the sharp bounds of the third Hankel determinant for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><msubsup><mi mathvariant="script">S</mi><mrow><mn>4</mn><mi mathvariant="script">L</mi></mrow><mo>*</mo></msubsup></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mrow><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac></semantics></math></inline-formula> bounded by a four-leaf shaped domain under the unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>.