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In the present paper, we discuss a class of bi-univalent analytic functions by applying a principle of differential subordinations and convolutions. We also formulate a class of bi-univalent functions influenced by a definition of a fractional <i>q</i>-derivative operator in an open symmetric unit disc. Further, we provide an estimate for the function 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> and <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> of the new classes. Over and above, we study an interesting Fekete–Szego inequality for each function in the newly defined classes.