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Article

Application on Fuzzy Third-Order Subordination and Superordination Connected with Lommel Function

by
Ekram E. Ali
1,2,
Georgia Irina Oros
3,*,
Rabha M. El-Ashwah
4 and
Abeer M. Albalahi
1
1
Department of Mathematics, Faculty 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
Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania
4
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(12), 1917; https://doi.org/10.3390/math13121917
Submission received: 4 April 2025 / Revised: 3 June 2025 / Accepted: 5 June 2025 / Published: 8 June 2025
(This article belongs to the Special Issue Current Topics in Geometric Function Theory, 2nd Edition)

Abstract

This work is based on the recently introduced concepts of third-order fuzzy differential subordination and its dual, third-order fuzzy differential superordination. In order to obtain the new results that add to the development of the newly initiated lines of research, a new operator is defined here using the concept of convolution and the normalized Lommel function. The methods focusing on the basic concept of admissible function are employed. Hence, the investigation of new third-order fuzzy differential subordination results starts with the definition of the suitable class of admissible functions. The first theorems discuss third-order fuzzy differential subordinations involving the newly introduced operator. The following result shows the conditions needed such that the fuzzy best dominant can be found for a third-order fuzzy differential subordination. Next, dual results are obtained by employing the methods of third-order fuzzy differential superordination based on the same concept of an admissible function. A suitable class of admissible functions is introduced and new third-order fuzzy differential superordinations are obtained, showing how the best subordinant can be obtained under certain restrictions. As a conclusion of this study, sandwhich-type results are derived, linking the outcome of the two dual fuzzy theories.
Keywords: fuzzy set; third-order fuzzy differential subordination; third-order fuzzy differential superordination; fuzzy dominant; fuzzy best dominant; fuzzy subordinant; best fuzzy subordinant; admissible function; Lommel function fuzzy set; third-order fuzzy differential subordination; third-order fuzzy differential superordination; fuzzy dominant; fuzzy best dominant; fuzzy subordinant; best fuzzy subordinant; admissible function; Lommel function

Share and Cite

MDPI and ACS Style

Ali, E.E.; Oros, G.I.; El-Ashwah, R.M.; Albalahi, A.M. Application on Fuzzy Third-Order Subordination and Superordination Connected with Lommel Function. Mathematics 2025, 13, 1917. https://doi.org/10.3390/math13121917

AMA Style

Ali EE, Oros GI, El-Ashwah RM, Albalahi AM. Application on Fuzzy Third-Order Subordination and Superordination Connected with Lommel Function. Mathematics. 2025; 13(12):1917. https://doi.org/10.3390/math13121917

Chicago/Turabian Style

Ali, Ekram E., Georgia Irina Oros, Rabha M. El-Ashwah, and Abeer M. Albalahi. 2025. "Application on Fuzzy Third-Order Subordination and Superordination Connected with Lommel Function" Mathematics 13, no. 12: 1917. https://doi.org/10.3390/math13121917

APA Style

Ali, E. E., Oros, G. I., El-Ashwah, R. M., & Albalahi, A. M. (2025). Application on Fuzzy Third-Order Subordination and Superordination Connected with Lommel Function. Mathematics, 13(12), 1917. https://doi.org/10.3390/math13121917

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