{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:54:49Z","timestamp":1760057689476,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,15]],"date-time":"2025-02-15T00:00:00Z","timestamp":1739577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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,\u03c1s(\u03bd,\u03c2) 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.<\/jats:p>","DOI":"10.3390\/axioms14020138","type":"journal-article","created":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T03:41:47Z","timestamp":1739763707000},"page":"138","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Application of Fuzzy Subordinations and Superordinations for an Analytic Function Connected with q-Difference Operator"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5477-0065","authenticated-orcid":false,"given":"Ekram E.","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 81451, Saudi Arabia"},{"name":"Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said 42521, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5490-3745","authenticated-orcid":false,"given":"Rabha M.","family":"El-Ashwah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7619-6869","authenticated-orcid":false,"given":"Abeer M.","family":"Albalahi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 81451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_2","first-page":"97","article-title":"The notion of subordination in fuzzy sets theory","volume":"19","author":"Oros","year":"2011","journal-title":"Gen. Math."},{"key":"ref_3","first-page":"55","article-title":"Fuzzy differential subordination","volume":"3","author":"Oros","year":"2012","journal-title":"Acta Univ. Apulensis"},{"key":"ref_4","first-page":"239","article-title":"Dominant and best dominant for fuzzy differential subordinations","volume":"57","author":"Oros","year":"2012","journal-title":"Stud. Univ. Babes-Bolyai Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Miller, S.S., and Mocanu, P.T. (2000). Differential Subordination Theory and Applications, Marcel Dekker.","DOI":"10.1201\/9781482289817"},{"key":"ref_6","first-page":"27","article-title":"Fuzzy Differential Superordination","volume":"7","author":"Atshan","year":"2017","journal-title":"Theory Appl. Math. Comput. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Oros, G.I. (2021). Fuzzy differential subordinations obtained using a hypergeometric integral operator. Mathematics, 20.","DOI":"10.3390\/math9202539"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1478","DOI":"10.55730\/1300-0098.3174","article-title":"Univalence criteria for analytic functions obtained using fuzzy differential subordinations","volume":"46","author":"Oros","year":"2022","journal-title":"Turk. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ali, E.E., Vivas-Cortez, M.J., El-Ashwah, R.M., and Albalahi, A.M. (2024). Fuzzy Subordination Results for Meromorphic Functions Connected with a Linear Operator. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8060308"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ali, E.E., Vivas-Cortez, M.J., and El-Ashwah, R.M. (2024). Fuzzy Differential Subordination for Classes of Admissible Functions Defined by a Class of Operators. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8070405"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5451","DOI":"10.3934\/math.2024263","article-title":"New results about fuzzy \u03b3-convex functions connected with the q-analogue multiplier-Noor integral operator","volume":"9","author":"Ali","year":"2024","journal-title":"AIMS Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"15569","DOI":"10.3934\/math.2023794","article-title":"Fuzzy differential subordination and superordination results for q-analogue of multiplier transformation","volume":"8","author":"Shah","year":"2023","journal-title":"AIMS Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"21053","DOI":"10.3934\/math.20241023","article-title":"Fuzzy differential subordination and superordination results for the Mittag-Leffler type Pascal distribution","volume":"9","author":"Soren","year":"2024","journal-title":"AIMS Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"18143","DOI":"10.3934\/math.2024886","article-title":"Subordinations and superordinations studies using q-difference operator","volume":"9","author":"Ali","year":"2024","journal-title":"Aims Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0080456800002751","article-title":"On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1909","journal-title":"Earth Environ. Sci. Trans. R. Soc. Edinb."},{"key":"ref_16","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Quart. J. Pure Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.1007\/s13370-021-00896-4","article-title":"Subordination factor sequence results for starlike and convex classes defined by q-Catas operator","volume":"32","author":"Aouf","year":"2021","journal-title":"Afr. Mat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"958563","DOI":"10.1155\/2014\/958563","article-title":"Some subordination results on q-analogue of Ruscheweyh differential operator","volume":"2014","author":"Aldweby","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_19","first-page":"77","article-title":"A generalization of starlike functions","volume":"14","author":"Ismail","year":"1990","journal-title":"Complex Var. Theory Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.2478\/s12175-014-0268-9","article-title":"Some classes of analytic functions related to conic domains","volume":"64","author":"Kanas","year":"2014","journal-title":"Math. Slovaca"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/s10476-017-0206-5","article-title":"On a class of analytic functions related to conic domains involving q-calculus","volume":"43","author":"Govindaraj","year":"2017","journal-title":"Anal. Math."},{"key":"ref_22","first-page":"1","article-title":"A certain q-Ruscheweyh type derivative operator and its applications involving multivalent functions","volume":"279","author":"Khan","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_23","first-page":"119","article-title":"On special fuzzy differential subordinations using Salagean and Ruscheweyh operators","volume":"261","author":"Oros","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_24","unstructured":"Whittaker, E.T., and Watson, G.N. (1927). A Course on Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, Cambridge University Press. [Fourth Edition (Reprinted)]."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.15672\/hujms.1319541","article-title":"Introduction in third-order fuzzy differential subordination","volume":"53","author":"Oros","year":"2004","journal-title":"Hacettepe J. Math. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Oros, G.I., Dzitac, S., and Bardac-Vlada, D.A. (2024). Introducing the Third-Order Fuzzy Superordination Concept and Related Results. Mathematics, 12.","DOI":"10.3390\/math12193095"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/138\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:35:02Z","timestamp":1760027702000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/138"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,15]]},"references-count":26,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["axioms14020138"],"URL":"https:\/\/doi.org\/10.3390\/axioms14020138","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,15]]}}}