| Summary: | In this research paper, we utilize the <i>q</i>-derivative concept to formulate specific differential and integral operators denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">R</mi><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>λ</mi></mrow></msubsup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>F</mi><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>λ</mi></mrow></msubsup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>G</mi><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>λ</mi></mrow></msubsup></semantics></math></inline-formula>. These operators are introduced with the aim of generalizing the class of Ruscheweyh operators within the set of univalent functions. We extract certain properties and characteristics of the set of differential subordinations employing specific techniques. By utilizing the newly defined operators, this paper goes on to establish subclasses of analytic functions defined on an open unit disc. Additionally, we delve into the convexity properties of the two recently introduced <i>q</i>-integral operators, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>F</mi><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>λ</mi></mrow></msubsup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>G</mi><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>λ</mi></mrow></msubsup></semantics></math></inline-formula>. Special cases of the primary findings are also discussed.
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