| Summary: | The aim of this paper is to introduce and investigate some new subclasses of bi-concave functions using <i>q</i>-convolution and some applications. These special cases are obtaining by making use of a <i>q</i>- derivative linear operator. For the new introduced subclasses, the authors obtain the first two initial Taylor–Maclaurin coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>c</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>c</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> of bi-concave functions. For certain values of the parameters, the authors deduce interesting corollaries for coefficient bounds which imply special cases of the new introduced operator. Also, we develop two examples for coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>c</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>c</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> for certain functions.
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