{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T18:33:02Z","timestamp":1781893982889,"version":"3.54.5"},"reference-count":52,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T00:00:00Z","timestamp":1684368000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Umm Al-Qura University","award":["23UQU4320576DSR002"],"award-info":[{"award-number":["23UQU4320576DSR002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the real world, there are many applications that find the Pascal distribution to be a useful and relevant model. One of these is the normal distribution. In this work, we develop a new subclass of analytic bi-univalent functions by making use of the q-Pascal distribution series as a construction. These functions involve the q-Gegenbauer polynomials, and we use them to establish our new subclass. Moreover, we solve the Fekete\u2013Szeg\u00f6 functional problem and analyze various different estimates of the Maclaurin coefficients for functions that belong to the new subclass.<\/jats:p>","DOI":"10.3390\/sym15051109","type":"journal-article","created":{"date-parts":[[2023,5,19]],"date-time":"2023-05-19T00:55:29Z","timestamp":1684457729000},"page":"1109","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Investigating New Subclasses of Bi-Univalent Functions Associated with q-Pascal Distribution Series Using the Subordination Principle"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5183-2654","authenticated-orcid":false,"given":"Abdullah","family":"Alsoboh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24382, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9287-7704","authenticated-orcid":false,"given":"Ala","family":"Amourah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9138-916X","authenticated-orcid":false,"given":"Maslina","family":"Darus","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8975-4102","authenticated-orcid":false,"given":"Carla Amoi","family":"Rudder","sequence":"additional","affiliation":[{"name":"Faculty of Resilience, Rabdan Academy, Abu Dhabi 22401, United Arab Emirates"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bain, L., and Engelhardt, M. (1992). Introduction to Probability and Mathematical Statistics, Duxburry Press.","DOI":"10.2307\/2532587"},{"key":"ref_2","first-page":"411","article-title":"Recherches sur laattraction des sph\u00e9roides homog\u00e9nes","volume":"10","author":"Legendre","year":"1785","journal-title":"M\u00c9Moires Pr\u00c9Sentes Par Divers. Savants Laacad\u00c9Mie Des Sci. Lainstitut Fr. Paris"},{"key":"ref_3","unstructured":"Bateman, H. (1953). Higher Transcendental Functions, McGRAW-HILL Book Company."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/S0377-0427(02)00642-8","article-title":"The Gegenbauer polynomials and typically real functions","volume":"153","author":"Kiepiela","year":"2003","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","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":"Trans. R. Soc. Edinb."},{"key":"ref_6","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1016\/j.na.2009.07.052","article-title":"On the Durrmeyer type modification of the q-baskakov type operators","volume":"72","author":"Aral","year":"2010","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"619","DOI":"10.2478\/s12175-011-0032-3","article-title":"Generalized q-Baskakov operators","volume":"61","author":"Aral","year":"2011","journal-title":"Math. Slovaca"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.endm.2018.05.005","article-title":"A Note On q-Integral Operators","volume":"67","author":"Aldweby","year":"2018","journal-title":"Electron. Notes Discret. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.2298\/FIL2005431A","article-title":"Connecting quantum calculus and harmonic starlike functions","volume":"34","author":"Ahuja","year":"2020","journal-title":"Filomat"},{"key":"ref_11","first-page":"8459405","article-title":"On Fekete-Szeg\u00f6 Problems for Certain Subclasses of Analytic Functions Defined by Differential Operator Involving q-Ruscheweyh Operator","volume":"2020","author":"Alsoboh","year":"2020","journal-title":"J. Funct. Space"},{"key":"ref_12","first-page":"761","article-title":"New subclass of analytic functions defined by q-differential operator with respect to k-symmetric points","volume":"1","author":"Alsoboh","year":"2019","journal-title":"Int. J. Math. Comput. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Gasper, G., and Rahman, M. (2004). Basic Hypergeometric Series, Cambridge University Press.","DOI":"10.1017\/CBO9780511526251"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Aral, A., Gupta, V., and Agarwal, R. (2013). Applications of q-Calculus in Operator Theory, Springer.","DOI":"10.1007\/978-1-4614-6946-9"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kumar, D., Ayant, F., S\u00fcdland, N., and Choi, J. (2023). Certain q-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function. Axioms, 12.","DOI":"10.3390\/axioms12010051"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1023\/B:IJTP.0000028885.42890.f5","article-title":"Disentangling q-exponentials: A general approach","volume":"43","author":"Quesne","year":"2004","journal-title":"Int. J. Theor. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0022-247X(78)90181-6","article-title":"Second Order Differential Inequalities in the Complex Plane","volume":"65","author":"Miller","year":"1978","journal-title":"J. Math. Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1307\/mmj\/1029002507","article-title":"Differential Subordinations and Univalent Functions","volume":"28","author":"Miller","year":"1981","journal-title":"Mich. Math. J."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Miller, S.S., and Mocanu, P.T. (2000). Differential Subordinations: Theory and Applications, Marcel Dekker, Inc.","DOI":"10.1201\/9781482289817"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1090\/S0002-9939-1967-0206255-1","article-title":"On a coefficient problem for bi-univalent functions","volume":"18","author":"Lewin","year":"1967","journal-title":"Proc. Amer. Math. Soc."},{"key":"ref_21","unstructured":"Brannan, D.A., and Clunie, J.G. (1980). Aspects of Contemporary Complex Analysis (Proceedings of the NATO Advanced Study Institute Held at the University of Durham, Durham; July 120, 1979), Academic Press."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/BF00247676","article-title":"The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |\u03be|<1","volume":"32","author":"Netanyahu","year":"1969","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Charalambides, C.A., and Papadatos, N. (2005). The q-factorial moments of discrete q-distributions and a characterization of the Euler distribution. Adv. Model. Charact. Appl., 57\u201371.","DOI":"10.1201\/9781420028690.ch3"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Askey, R., and Ismail, M.E.H. (1983). A Generalization of Ultraspherical Polynomials, Studies of Pure Mathematics, Birkhauser.","DOI":"10.1007\/978-3-0348-5438-2_6"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"12371","DOI":"10.1088\/0305-4470\/39\/40\/006","article-title":"New connection formulae for the q\u2013orthogonal polynomials via a series expansion of the q\u2013exponential","volume":"39","author":"Chakrabarti","year":"2006","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_26","first-page":"5574673","article-title":"Fekete-Szeg\u00f6 inequality for analytic and bi-univalent functions subordinate to Gegenbauer polynomials","volume":"2021","author":"Amourah","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_27","first-page":"625","article-title":"Gegenbauer polynomials and bi-univalent functions","volume":"10","author":"Amourah","year":"2021","journal-title":"Palest. J. Math."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Amourah, A., Alsoboh, A., Ogilat, O., Gharib, G.M., Saadeh, R., and Al Soudi, M. (2023). A Generalization of Gegenbauer Polynomials and Bi-Univalent Functions. Axioms, 12.","DOI":"10.3390\/axioms12020128"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Alsoboh, A., Amourah, A., Darus, M., and Sharefeen, R.I.A. (2023). Applications of Neutrosophic q-Poisson distribution Series for Subclass of Analytic Functions and Bi-Univalent Functions. Mathematics, 11.","DOI":"10.3390\/math11040868"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Amourah, A., Frasin, B.A., and Seoudy, T.M. (2022). An Application of Miller\u2013Ross-Type Poisson Distribution on Certain Subclasses of Bi-Univalent Functions Subordinate to Gegenbauer Polynomials. Mathematics, 10.","DOI":"10.3390\/math10142462"},{"key":"ref_31","first-page":"11","article-title":"An application of binomial distribution series on certain analytic functions","volume":"62","author":"Nazeer","year":"2019","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_32","first-page":"012003","article-title":"On Fekete-Szego problem associated with q-derivative operator","volume":"1212","author":"Alsoboh","year":"2019","journal-title":"J. Phys."},{"key":"ref_33","first-page":"693908","article-title":"Coefficient estimate of bi-univalent functions of complex order associated with the Hohlov operator","volume":"2014","author":"Peng","year":"2014","journal-title":"Complex Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.aml.2010.05.009","article-title":"Certain subclasses of analytic and bi-univalent functions","volume":"23","author":"Srivastava","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Buyankara, M., \u00c7a\u011flar, M., and Cot\u00eerl\u0103, L.-I. (2022). New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials. Axioms, 11.","DOI":"10.3390\/axioms11110652"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1080\/27690911.2022.2088743","article-title":"Applications of q-derivative operator to subclasses of bi-univalent functions involving Gegenbauer polynomials","volume":"30","author":"Hu","year":"2022","journal-title":"Appl. Math. Sci. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"\u00c7a\u011flar, M., Cot\u00eerl\u0103, L.-I., and Buyankara, M. (2022). Fekete\u2013Szeg\u00f6 Inequalities for a New Subclass of Bi-Univalent Functions Associated with Gegenbauer Polynomials. Symmetry, 14.","DOI":"10.3390\/sym14081572"},{"key":"ref_38","first-page":"137","article-title":"On Fekete-Szeg\u00f6 problem for a new subclass of bi-univalent functions defined by Bernoulli polynomials","volume":"71","author":"Buyankara","year":"2022","journal-title":"Acta Univ. Apulensis"},{"key":"ref_39","first-page":"123","article-title":"On subclasses of harmonic univalent functions defined by Jackson derivative","volume":"10","author":"Alsoboh","year":"2019","journal-title":"J. Anal."},{"key":"ref_40","first-page":"1608","article-title":"Chebyshev polynomial coefficient bounds for a subclass of bi-univalent functions","volume":"72","year":"2019","journal-title":"CR Acad. Bulgare Sci."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Venkateswarlu, B., Thirupathi Reddy, P., Alt\u0131nkaya, \u015e., Boonsatit, N., Hammachukiattikul, P., and Sujatha, V. (2022). On a Certain Subclass of Analytic Functions Defined by Touchard Polynomials. Symmetry, 14.","DOI":"10.3390\/sym14040838"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Al-Hawary, T., Amourah, A., Alsoboh, A., and Alsalhi, O. (2023). A New Comprehensive Subclass of Analytic Bi-Univalent Functions Related to Gegenbauer Polynomials. Symmetry, 15.","DOI":"10.3390\/sym15030576"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"49","DOI":"10.7153\/jca-02-05","article-title":"Certain subclasses of bi\u2013univalent functions satisfying subordinate conditions","volume":"2","author":"Deniz","year":"2013","journal-title":"J. Classical Anal."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.15672\/hujms.557072","article-title":"Fekete-Szeg\u00f6 problem for generalized bi-subordinate functions of complex order","volume":"49","author":"Deniz","year":"2020","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"7259","DOI":"10.3934\/math.2020464","article-title":"A certain subclass of bi-univalent functions associated with Bell numbers and q-Srivastava Attiya operator","volume":"5","author":"Deniz","year":"2020","journal-title":"AIMS Math."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/s40590-019-00265-z","article-title":"Generalized distribution for analytic function classes associated with error functions and Bell numbers","volume":"26","author":"Oluwafemi","year":"2020","journal-title":"Bolet\u00edn Soc. Matem\u00e1tica Mex."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Amourah, A., Frasin, B.A., Ahmad, M., and Yousef, F. (2022). Exploiting the Pascal Distribution Series and Gegenbauer Polynomials to Construct and Study a New Subclass of Analytic Bi-Univalent Functions. Symmetry, 14.","DOI":"10.3390\/sym14010147"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"8355285","DOI":"10.1155\/2022\/8355285","article-title":"Subclass of Analytic Functions Related with Pascal Distribution Series","volume":"2022","author":"Shammaky","year":"2022","journal-title":"J. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"6354994","DOI":"10.1155\/2022\/6354994","article-title":"Consolidation of a Certain Discrete Probability Distribution with a Subclass of Bi-Univalent Functions Involving Gegenbauer Polynomials","volume":"2022","author":"Amourah","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"71","DOI":"10.34198\/ejms.4120.7182","article-title":"Applications of Borel distribution series on analytic functions","volume":"4","author":"Wanas","year":"2020","journal-title":"Earthline J. Math. Sci."},{"key":"ref_51","unstructured":"Pommerenke, C. (1975). Studia Mathematica Mathematische Lehrbucher, Vandenhoeck and Ruprecht."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"169","DOI":"10.36045\/bbms\/1394544302","article-title":"On the Fekete-Szeg\u00f6 problem for classes of bi-univalent functions","volume":"21","author":"Zaprawa","year":"2014","journal-title":"Bull. Belg. Math.-Soc.-Simon Stevin"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1109\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:38:02Z","timestamp":1760125082000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1109"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,18]]},"references-count":52,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["sym15051109"],"URL":"https:\/\/doi.org\/10.3390\/sym15051109","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,18]]}}}