Fuzzy Differential Subordination Associated with a General Linear Transformation

Sarfraz Nawaz Malik, Nazar Khan, Ferdous M. O. Tawfiq, Mohammad Faisal Khan, Qazi Zahoor Ahmad, Qin Xin

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Issue number22
Publication statusPublished - 8 Nov 2023


  • linear transformation
  • fuzzy differential subordination
  • fuzzy set
  • analytic functions
  • Al-Oboudi differential operator
  • Babalola convolution operator


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