{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T15:11:56Z","timestamp":1762182716161,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T00:00:00Z","timestamp":1762041600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University","award":["IMSIU-DDRSP2504"],"award-info":[{"award-number":["IMSIU-DDRSP2504"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study introduces a new integral operator Ip,\u03bc\u03b4 that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses: \u03a3p+\u03b4,\u03bc,\u03b1, consisting of functions with nonnegative coefficients, and \u03a3p+\u03b4,\u03bc,\u03b1,c, which further fixes the second positive coefficient. For these classes, we establish a necessary and sufficient coefficient condition, which serves as the foundation for deriving a set of sharp results. These include accurate coefficient bounds, distortion theorems for functions and derivatives, and radii of starlikeness and convexity of a specific order. Furthermore, we demonstrate the closure property of the class \u03a3p+\u03b4,\u03bc,\u03b1,c, identify its extreme points, and then construct a neighborhood theorem. All the findings presented in this paper are sharp. To demonstrate the practical utility of our symmetric operator paradigm, we apply it to a canonical fractional electrodynamics problem. We demonstrate how sharp distortion theorems establish rigorous, time-invariant upper bounds for a solitary electrostatic potential and its accompanying electric field, resulting in a mathematically guaranteed safety buffer against dielectric breakdown. This study develops a symmetric and consistent approach to investigating the geometric characteristics of meromorphic multivalent functions and their applications in physical models.<\/jats:p>","DOI":"10.3390\/sym17111837","type":"journal-article","created":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T13:47:58Z","timestamp":1762177678000},"page":"1837","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Generalization of the Rafid Operator and Its Symmetric Role in Meromorphic Function Theory with Electrostatic Applications"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-9337-7221","authenticated-orcid":false,"given":"Aya F.","family":"Elkhatib","sequence":"first","affiliation":[{"name":"Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Beni-Suef 62611, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7676-1020","authenticated-orcid":false,"given":"Atef F.","family":"Hashem","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adela O.","family":"Mostafa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansura University, Mansura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammed M.","family":"Tharwat","sequence":"additional","affiliation":[{"name":"Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Beni-Suef 62611, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,2]]},"reference":[{"key":"ref_1","first-page":"11","article-title":"Zur Theorie der eindeutigen analytischen Functionen","volume":"1876","author":"Weierstrass","year":"1876","journal-title":"Abh. 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