Application of Fuzzy Subordinations and Superordinations for an Analytic Function Connected with <inline-formula><math display="inline"><semantics><mstyle mathvariant="bold"><mrow><mi mathvariant="fraktur">q</mi></mrow></mstyle></semantics></math></inline-formula>-Difference Operator

Abstract

This paper extends the idea of subordination from the theory of fuzzy sets to the geometry theory of analytic functions with a single complex variable. The purpose of this work is to define fuzzy subordination and illustrate its main characteristics. New fuzzy differential subordinations will be introduced with the help of this effort. We define a linear operator Iq,ρs(ν,ς) using the concept of the q-calculus operators. New fuzzy differential subordinations are created by employing the previously described operator, functions from the new class, and well-known lemmas. Specific corollaries derived from the operator proved the many examples created for the fuzzy differential subordinations, as well as the theorems, and demonstrate how the new theoretical conclusions apply to the fuzzy differential superordinations provided in this research.

Keywords

• analytic function
• differential subordination
• superordination
• q -difference operator%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm10001"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">q</mml:mi> </mml:mrow> </mml:semantics> </mml:math></inline-formula></named-content>-difference operator
• q -analogue Catas operator%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm10002"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">q</mml:mi> </mml:mrow> </mml:semantics> </mml:math></inline-formula></named-content>-analogue Catas operator
• q -analogue of Ruscheweyh operator%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm10003"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">q</mml:mi> </mml:mrow> </mml:semantics> </mml:math></inline-formula></named-content>-analogue of Ruscheweyh operator