| Summary: | In this study, we investigate a possible relationship between fuzzy differential subordination and the theory of geometric functions. First, using the Al-Oboudi differential operator and the Babalola convolution operator, we establish the new operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">BS</mi><mrow><mi>α</mi><mo>,</mo><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>t</mi></mrow></msubsup><mo>:</mo></mrow></semantics></math></inline-formula>A<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>n</mi></msub><mspace width="3.33333pt"></mspace><mo>→</mo><mspace width="3.33333pt"></mspace></mrow></semantics></math></inline-formula>A<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mi>n</mi></msub></semantics></math></inline-formula> in the open unit disc <i>U</i>. The second step is to develop fuzzy differential subordination for the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">BS</mi><mrow><mi>α</mi><mo>,</mo><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>t</mi></mrow></msubsup></semantics></math></inline-formula>. By considering linear transformations of the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">BS</mi><mrow><mi>α</mi><mo>,</mo><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>t</mi></mrow></msubsup></semantics></math></inline-formula>, we define a new fuzzy class of analytic functions in <i>U</i> which we denote by T<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow></mrow><mrow><mi mathvariant="sans-serif-italic">ϝ</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>t</mi></mrow></msubsup><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>δ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Several innovative results are found using the concept of fuzzy differential subordination and the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">BS</mi><mrow><mi>α</mi><mo>,</mo><mi>λ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>t</mi></mrow></msubsup></semantics></math></inline-formula> for the function <i>f</i> in the class T<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow></mrow><mrow><mi mathvariant="sans-serif-italic">ϝ</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>t</mi></mrow></msubsup><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>δ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In addition, we explore a number of examples and corollaries to illustrate the implications of our key findings. Finally, we highlight several established results to demonstrate the connections between our work and existing studies.
|
|---|