Fuzzy Differential Subordination Associated with a General Linear Transformation

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 ℬ𝒮𝑚,𝑡𝛼,𝜆:A𝑛→A𝑛 in the open unit disc U. The second step is to develop fuzzy differential subordination for the operator ℬ𝒮𝑚,𝑡𝛼,𝜆. By considering linear transformations of the operator ℬ𝒮𝑚,𝑡𝛼,𝜆, we define a new fuzzy class of analytic functions in U which we denote by T𝜆,𝑡ϝ(𝑚,𝛼,𝛿). Several innovative results are found using the concept of fuzzy differential subordination and the operator ℬ𝒮𝑚,𝑡𝛼,𝜆 for the function f in the class T𝜆,𝑡ϝ(𝑚,𝛼,𝛿). 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.