Applications of fuzzy differential subordination theory on analytic $ p $ -valent functions connected with $ \mathfrak{q} $-calculus operator

Ekram E. Ali; Georgia Irina Oros; Rabha M. El-Ashwah; Abeer M. Albalahi; Ekram E. Ali; Georgia Irina Oros; Rabha M. El-Ashwah; Abeer M. Albalahi

doi:10.3934/math.20241031

AIMS Mathematics

2024, Volume 9, Issue 8: 21239-21254. doi: 10.3934/math.20241031

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Research article

Applications of fuzzy differential subordination theory on analytic $ p $ -valent functions connected with $ \mathfrak{q} $-calculus operator

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 81451, Saudi Arabia
2.
Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said 42521, Egypt
3.
University of Oradea, Department of Mathematics, Universitatii 1, 410087 Oradea, Romania
4.
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

Received: 27 April 2024 Revised: 07 June 2024 Accepted: 18 June 2024 Published: 01 July 2024
MSC : 30C45, 30C80

In recent years, the concept of fuzzy set has been incorporated into the field of geometric function theory, leading to the evolution of the classical concept of differential subordination into that of fuzzy differential subordination. In this study, certain generalized classes of $ p $ -valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy $ p $-valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a $ \mathfrak{q} $-calculus operator defined in this investigation using the concept of convolution. Some inclusion results are discussed concerning the newly introduced classes based on the means given by the fuzzy differential subordination theory. Furthermore, connections are shown between the important results of this investigation and earlier ones. The second part of the investigation concerns a new generalized $ \mathfrak{q} $-calculus operator, defined here and having the $ (p, \mathfrak{q)} $-Bernardi operator as particular case, applied to the functions belonging to the new classes introduced in this study. Connections between the classes are established through this operator.
Citation: Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi. Applications of fuzzy differential subordination theory on analytic $ p $ -valent functions connected with $ \mathfrak{q} $-calculus operator[J]. AIMS Mathematics, 2024, 9(8): 21239-21254. doi: 10.3934/math.20241031

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Abstract

In recent years, the concept of fuzzy set has been incorporated into the field of geometric function theory, leading to the evolution of the classical concept of differential subordination into that of fuzzy differential subordination. In this study, certain generalized classes of $ p $ -valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy $ p $-valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a $ \mathfrak{q} $-calculus operator defined in this investigation using the concept of convolution. Some inclusion results are discussed concerning the newly introduced classes based on the means given by the fuzzy differential subordination theory. Furthermore, connections are shown between the important results of this investigation and earlier ones. The second part of the investigation concerns a new generalized $ \mathfrak{q} $-calculus operator, defined here and having the $ (p, \mathfrak{q)} $-Bernardi operator as particular case, applied to the functions belonging to the new classes introduced in this study. Connections between the classes are established through this operator.

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