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Motivated by q-calculus, we define a new family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Σ</mo></semantics></math></inline-formula>, which is the family of bi-univalent analytic functions in the open unit disc <i>U</i> that is related to the Einstein function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We establish estimates for the first two Taylor–Maclaurin coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>2</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, and the Fekete–Szegö inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><msub><mi>a</mi><mn>3</mn></msub><mo>−</mo><mi>μ</mi><msubsup><mi>a</mi><mn>2</mn><mn>2</mn></msubsup></mfenced></semantics></math></inline-formula> for the functions that belong to these families.