{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T20:11:13Z","timestamp":1774296673724,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T00:00:00Z","timestamp":1665014400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we make use of the Riemann\u2013Liouville fractional q-integral operator to discuss the class Sq,\u03b4*(\u03b1) of univalent functions for \u03b4&gt;0,\u03b1\u2208C\u2212{0}, and 0&lt;|q|&lt;1. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional q-difference operator. Moreover, we derive the normalized classes P\u03b4,q\u03b6(\u03b2,\u03b3) and P\u03b4,q(\u03b2) (0&lt;|q|&lt;1, \u03b4\u22650,0\u2264\u03b2\u22641,\u03b6&gt;0) of analytic functions on a unit disc and provide conditions for the parameters q,\u03b4,\u03b6,\u03b2, and \u03b3 so that P\u03b4,q\u03b6(\u03b2,\u03b3)\u2282Sq,\u03b4*(\u03b1) and P\u03b4,q(\u03b2)\u2282Sq,\u03b4*(\u03b1) for \u03b1\u2208C\u2212{0}. Finally, we also propose an application to symmetric q-analogues and Ruscheweh\u2019s duality theory.<\/jats:p>","DOI":"10.3390\/sym14102076","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T03:32:56Z","timestamp":1665459176000},"page":"2076","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Duality on q-Starlike Functions Associated with Fractional q-Integral Operators and Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7100-1199","authenticated-orcid":false,"given":"Ebrahim","family":"Amini","sequence":"first","affiliation":[{"name":"Department of Mathematics, Payme Noor University, Tehran P.O. Box 19395-4697, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shrideh","family":"Al-Omari","sequence":"additional","affiliation":[{"name":"Department of Scientific Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7741-5322","authenticated-orcid":false,"given":"Mojtaba","family":"Fardi","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord P.O. Box 115, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2001). Quantum Calculus (Universitext). Softcover Reprint of the Original, Springer. [1st ed.].","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/S0898-1221(04)90025-9","article-title":"Integral inequalities in q-calculus","volume":"47","author":"Gauchman","year":"2004","journal-title":"Comput. Math. 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