| Summary: | We defined two new subclasses of analytic bi-univalent function class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Σ</mo><mo>,</mo></mrow></semantics></math></inline-formula> in the open unit disk related with the Sălăgean–Erdély–Kober operator. The bounds on initial coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">|</mo></mrow><msub><mi>a</mi><mn>2</mn></msub><mrow><mo stretchy="false">|</mo><mo>,</mo><mo stretchy="false">|</mo></mrow><msub><mi>a</mi><mn>3</mn></msub><mrow><mo stretchy="false">|</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 stretchy="false">|</mo></mrow><msub><mi>a</mi><mn>4</mn></msub><mrow><mo stretchy="false">|</mo></mrow></mrow></semantics></math></inline-formula> for the functions in these new subclasses of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Σ</mo></semantics></math></inline-formula> are investigated. Using the estimates of coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula>, we also discuss the Fekete-Szegö inequality results for the function classes defined in this paper. Relevant connections of these results, presented here as corollaries, are new and not studied in association with Sălăgean-Erdély–Kober operator for the subclasses defined earlier.
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