| Summary: | In this paper, we introduce a special family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="fraktur">M</mi><msub><mi>σ</mi><mi>m</mi></msub></msub><mrow><mo>(</mo><mi>τ</mi><mo>,</mo><mi>ν</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>φ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of the function family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>m</mi></msub></semantics></math></inline-formula> of <i>m</i>-fold symmetric bi-univalent functions defined in the open unit disc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">D</mi></semantics></math></inline-formula> and obtain estimates of the first two Taylor–Maclaurin coefficients for functions in the special family. Further, the Fekete–Szegö functional for functions in this special family is also estimated. The results presented in this paper not only generalize and improve some recent works, but also give new results as special cases.
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