{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T14:45:21Z","timestamp":1774363521710,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,17]],"date-time":"2020-02-17T00:00:00Z","timestamp":1581897600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, Henan, P.R. China","award":["School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, Henan, P.R. China"],"award-info":[{"award-number":["School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, Henan, P.R. China"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In analysis, the introduction of q-calculus has been a revelation. It has a deep impact on various concepts and applications of pure and applied sciences. In this article we investigate certain geometric properties relating to convolution of functions of a newly defined class of analytic functions. The important region of the lemniscate of Bernoulli is considered. Here we utilize concepts of q-calculus which enhances and generalizes the vitality of this research work. In the same context we study the Fekete\u2013Szeg\u00f6 problem.<\/jats:p>","DOI":"10.3390\/sym12020291","type":"journal-article","created":{"date-parts":[[2020,2,26]],"date-time":"2020-02-26T04:18:29Z","timestamp":1582690709000},"page":"291","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Some Geometric Properties of a Family of Analytic Functions Involving a Generalized q-Operator"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3079-9944","authenticated-orcid":false,"given":"Lei","family":"Shi","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, China"}]},{"given":"Muhammad","family":"Ghaffar Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3716-2818","authenticated-orcid":false,"given":"Bakhtiar","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Govt. Degree College Mardan, Mardan 23200, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,17]]},"reference":[{"key":"ref_1","first-page":"109","article-title":"On q-Baskakov type operators","volume":"42","author":"Aral","year":"2009","journal-title":"Demonstr. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"368","DOI":"10.3906\/mat-1601-34","article-title":"On certain GBS-Durrmeyer operators based on q-integers","volume":"41","author":"Barbosu","year":"2017","journal-title":"Turk. J. Math."},{"key":"ref_3","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_4","first-page":"253","article-title":"On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1909","journal-title":"Earth Environ. Sci. Trans. R. Edinb."},{"key":"ref_5","unstructured":"Srivastava, H.M., and Owa, S. (1989). Univalent functions, fractional calculus, and associated generalized hypergeometric functions. Univalent Functions, Fractional Calculus, and Their Applications, John Wiley and Sons."},{"key":"ref_6","first-page":"61","article-title":"Close-to-convexity of a certain family of q-Mittag-Leffler functions","volume":"1","author":"Srivastava","year":"2017","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"15","DOI":"10.14492\/hokmj\/1498788094","article-title":"A generalization of starlike functions of order \u03b1","volume":"46","author":"Agrawal","year":"2017","journal-title":"Hokkaido Math. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.2478\/s12175-014-0268-9","article-title":"Some class of analytic functions related to conic domains","volume":"64","author":"Kanas","year":"2014","journal-title":"Math. Slovaca."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"501","DOI":"10.12732\/ijam.v30i6.4","article-title":"A novel subclass of univalent functions involving operators of fractional calculus","volume":"30","author":"Kamble","year":"2017","journal-title":"Int. J. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1515\/ms-2017-0166","article-title":"New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator","volume":"68","author":"Arif","year":"2018","journal-title":"Math. Slovaca"},{"key":"ref_11","first-page":"75","article-title":"New subfamily of meromorphic convex functions in circular domain involving q-operator","volume":"16","author":"Ahmad","year":"2018","journal-title":"In. J. Anal. Appl."},{"key":"ref_12","first-page":"3818915","article-title":"A subfamily of univalent functions associated with q-analogue of Noor integral operator","volume":"2018","author":"Arif","year":"2018","journal-title":"J. Funct. Spaces"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1345","DOI":"10.1007\/s00025-016-0592-1","article-title":"New subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator","volume":"71","author":"Mahmmod","year":"2017","journal-title":"Results Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1186\/1029-242X-2013-412","article-title":"Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli","volume":"2013","author":"Raza","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_15","unstructured":"Li, Z., Ren, F., Yang, L., and Zhang, S. (1992, January 19\u201323). Unified treatment of some special classes of univalent functions. Proceedings of the Conference on Complex Analysis, Tianjin, China."},{"key":"ref_16","first-page":"1","article-title":"Fekete\u2013Szeg\u00f6 inequality for Certain Class of Analytic Functions","volume":"1","author":"Ravichandran","year":"2004","journal-title":"Far East. J. Math. Anal. Appl."},{"key":"ref_17","first-page":"171","article-title":"Fekete\u2013Szeg\u00f6 inequality for certain class of Bazilevic functions","volume":"15","author":"Ravichandran","year":"2004","journal-title":"Far East J. Math. Sci."},{"key":"ref_18","first-page":"9","article-title":"Convex functions of complex order","volume":"34","author":"Ravichandran","year":"2005","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_19","first-page":"29","article-title":"On Certain Subclasses of a new class of analytic functions","volume":"28","author":"Shanmugam","year":"2007","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_20","first-page":"283","article-title":"Fekete\u2013Szeg\u00f6 inequality for certain class of Bazilevic functions","volume":"34","author":"Shanmugam","year":"2006","journal-title":"Int. Math."},{"key":"ref_21","first-page":"383","article-title":"An application of the fractional derivative","volume":"29","author":"Srivastava","year":"1984","journal-title":"Math. Jpn."},{"key":"ref_22","first-page":"101","article-title":"Radius of convexity of some subclasses of strongly starlike functions","volume":"19","author":"Stankiewicz","year":"1996","journal-title":"Zesz. Nauk. Politech. Rzesz. Mater."},{"key":"ref_23","first-page":"569","article-title":"Radius problem in the class SL*","volume":"214","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"93","DOI":"10.22342\/jims.18.2.115.93-99","article-title":"Applications of certain functions associated with lemniscate Bernoulli","volume":"18","author":"Halim","year":"2012","journal-title":"J. Indones. Math. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.11650\/twjm\/1500406676","article-title":"First order differential subordination for functions associated with the lemniscate of Bernoulli","volume":"16","author":"Ali","year":"2012","journal-title":"Taiwan. J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"349","DOI":"10.5666\/KMJ.2009.49.2.349","article-title":"Coefficient estimates in a class of strongly starlike functions","volume":"49","year":"2009","journal-title":"Kyungpook Math. J."},{"key":"ref_27","first-page":"454","article-title":"On new classes of integral operators","volume":"65","author":"Noor","year":"2013","journal-title":"J. Nat. Geom."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1006\/jmaa.1999.6501","article-title":"On integral operators","volume":"238","author":"Noor","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"699","DOI":"10.11568\/kjm.2014.22.4.699","article-title":"Starlikeness of q-differential operator involving quantum calculus","volume":"22","author":"Aldawish","year":"2014","journal-title":"Korean J. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2013\/382312","article-title":"A subclass of harmonic univalent functions associated with q-analogue of Dziok-Srivastava operator","volume":"2013","author":"Aldweby","year":"2013","journal-title":"ISRN Math. Anal."},{"key":"ref_31","first-page":"454","article-title":"A generalized operator involving the q-hypergeometric function","volume":"65","author":"Mohammed","year":"2013","journal-title":"Matemati\u010dki Vesnik"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"135","DOI":"10.7153\/jmi-10-11","article-title":"Coefficient estimates of new classes of q-starlike and q-convex functions of complex order","volume":"10","author":"Seoudy","year":"2016","journal-title":"J. Math. 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