{"id":"https://openalex.org/W4318473662","doi":"https://doi.org/10.3390/axioms12020128","title":"A Generalization of Gegenbauer Polynomials and Bi-Univalent Functions","display_name":"A Generalization of Gegenbauer Polynomials and Bi-Univalent Functions","publication_year":2023,"publication_date":"2023-01-28","ids":{"openalex":"https://openalex.org/W4318473662","doi":"https://doi.org/10.3390/axioms12020128"},"language":"en","primary_location":{"id":"doi:10.3390/axioms12020128","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12020128","pdf_url":"https://www.mdpi.com/2075-1680/12/2/128/pdf?version=1675217924","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/12/2/128/pdf?version=1675217924","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091401373","display_name":"Ala Amourah","orcid":"https://orcid.org/0000-0001-9287-7704"},"institutions":[{"id":"https://openalex.org/I97864281","display_name":"Irbid National University","ror":"https://ror.org/00xfxvy87","country_code":"JO","type":"education","lineage":["https://openalex.org/I97864281"]}],"countries":["JO"],"is_corresponding":true,"raw_author_name":"Ala Amourah","raw_affiliation_strings":["Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan"],"raw_orcid":"https://orcid.org/0000-0001-9287-7704","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan","institution_ids":["https://openalex.org/I97864281"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083883489","display_name":"Abdullah Alsoboh","orcid":"https://orcid.org/0000-0002-5183-2654"},"institutions":[{"id":"https://openalex.org/I199693650","display_name":"Umm al-Qura University","ror":"https://ror.org/01xjqrm90","country_code":"SA","type":"education","lineage":["https://openalex.org/I199693650"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Abdullah Alsoboh","raw_affiliation_strings":["Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24231, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-5183-2654","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24231, Saudi Arabia","institution_ids":["https://openalex.org/I199693650"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5023592096","display_name":"Osama Ogilat","orcid":"https://orcid.org/0000-0003-2370-6332"},"institutions":[{"id":"https://openalex.org/I104941359","display_name":"Al-Ahliyya Amman University","ror":"https://ror.org/00xddhq60","country_code":"JO","type":"education","lineage":["https://openalex.org/I104941359"]}],"countries":["JO"],"is_corresponding":false,"raw_author_name":"Osama Ogilat","raw_affiliation_strings":["Department of Basic Sciences, Faculty of Arts and Science, Al-Ahliyya Amman University, Amman 19328, Jordan"],"raw_orcid":"https://orcid.org/0000-0003-2370-6332","affiliations":[{"raw_affiliation_string":"Department of Basic Sciences, Faculty of Arts and Science, Al-Ahliyya Amman University, Amman 19328, Jordan","institution_ids":["https://openalex.org/I104941359"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013519661","display_name":"Gharib Mousa Gharib","orcid":"https://orcid.org/0000-0001-8645-1029"},"institutions":[{"id":"https://openalex.org/I153687341","display_name":"Zarqa University","ror":"https://ror.org/01wf1es90","country_code":"JO","type":"education","lineage":["https://openalex.org/I153687341"]}],"countries":["JO"],"is_corresponding":false,"raw_author_name":"Gharib Mousa Gharib","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan"],"raw_orcid":"https://orcid.org/0000-0001-8645-1029","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan","institution_ids":["https://openalex.org/I153687341"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025732055","display_name":"Rania Saadeh","orcid":"https://orcid.org/0000-0002-6394-1452"},"institutions":[{"id":"https://openalex.org/I153687341","display_name":"Zarqa University","ror":"https://ror.org/01wf1es90","country_code":"JO","type":"education","lineage":["https://openalex.org/I153687341"]}],"countries":["JO"],"is_corresponding":false,"raw_author_name":"Rania Saadeh","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan"],"raw_orcid":"https://orcid.org/0000-0002-6394-1452","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan","institution_ids":["https://openalex.org/I153687341"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089688788","display_name":"Maha Al Soudi","orcid":null},"institutions":[{"id":"https://openalex.org/I146367977","display_name":"Applied Science Private University","ror":"https://ror.org/01ah6nb52","country_code":"JO","type":"education","lineage":["https://openalex.org/I146367977"]}],"countries":["JO"],"is_corresponding":false,"raw_author_name":"Maha Al Soudi","raw_affiliation_strings":["Department of Basic Scientific Sciences, Applied Science Private University, Amman 11931, Jordan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Basic Scientific Sciences, Applied Science Private University, Amman 11931, Jordan","institution_ids":["https://openalex.org/I146367977"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5083883489","https://openalex.org/A5091401373"],"corresponding_institution_ids":["https://openalex.org/I199693650","https://openalex.org/I97864281"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":17.8756,"has_fulltext":true,"cited_by_count":37,"citation_normalized_percentile":{"value":0.9980572,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":100},"biblio":{"volume":"12","issue":"2","first_page":"128","last_page":"128"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11822","display_name":"Analytic and geometric function theory","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11822","display_name":"Analytic and geometric function theory","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9577999711036682,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9488999843597412,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/gegenbauer-polynomials","display_name":"Gegenbauer polynomials","score":0.7842503190040588},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.7606570720672607},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6477170586585999},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.6141377687454224},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.5625060200691223},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5496245622634888},{"id":"https://openalex.org/keywords/difference-polynomials","display_name":"Difference polynomials","score":0.5391964912414551},{"id":"https://openalex.org/keywords/discrete-orthogonal-polynomials","display_name":"Discrete orthogonal polynomials","score":0.4979054927825928},{"id":"https://openalex.org/keywords/wilson-polynomials","display_name":"Wilson polynomials","score":0.4910705089569092},{"id":"https://openalex.org/keywords/jacobi-polynomials","display_name":"Jacobi polynomials","score":0.48445940017700195},{"id":"https://openalex.org/keywords/hahn-polynomials","display_name":"Hahn polynomials","score":0.4545142650604248},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3611752986907959},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.31152909994125366}],"concepts":[{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.7842503190040588},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.7606570720672607},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6477170586585999},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.6141377687454224},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.5625060200691223},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5496245622634888},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.5391964912414551},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.4979054927825928},{"id":"https://openalex.org/C43176163","wikidata":"https://www.wikidata.org/wiki/Q8023263","display_name":"Wilson polynomials","level":4,"score":0.4910705089569092},{"id":"https://openalex.org/C48490523","wikidata":"https://www.wikidata.org/wiki/Q371631","display_name":"Jacobi polynomials","level":3,"score":0.48445940017700195},{"id":"https://openalex.org/C61349858","wikidata":"https://www.wikidata.org/wiki/Q1568814","display_name":"Hahn polynomials","level":5,"score":0.4545142650604248},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3611752986907959},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31152909994125366}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms12020128","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12020128","pdf_url":"https://www.mdpi.com/2075-1680/12/2/128/pdf?version=1675217924","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:5de25e0fd6014ebc9af1d4d41e68e1bf","is_oa":true,"landing_page_url":"https://doaj.org/article/5de25e0fd6014ebc9af1d4d41e68e1bf","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 12, Iss 2, p 128 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/12/2/128/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms12020128","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms; Volume 12; Issue 2; Pages: 128","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms12020128","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12020128","pdf_url":"https://www.mdpi.com/2075-1680/12/2/128/pdf?version=1675217924","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.5899999737739563,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4318473662.pdf"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W1963790780","https://openalex.org/W1988930470","https://openalex.org/W2039692239","https://openalex.org/W2043190453","https://openalex.org/W2043209803","https://openalex.org/W2124376566","https://openalex.org/W2134197866","https://openalex.org/W2170944871","https://openalex.org/W2562512100","https://openalex.org/W2735956175","https://openalex.org/W2761672173","https://openalex.org/W2787133192","https://openalex.org/W2885198535","https://openalex.org/W2894519081","https://openalex.org/W2954358791","https://openalex.org/W3024724767","https://openalex.org/W3093697257","https://openalex.org/W3128248869","https://openalex.org/W4281807159","https://openalex.org/W4296231548","https://openalex.org/W6684907877","https://openalex.org/W6741100133","https://openalex.org/W6766538240","https://openalex.org/W6843416246"],"related_works":["https://openalex.org/W2072647369","https://openalex.org/W2070537393","https://openalex.org/W2132314122","https://openalex.org/W2045154321","https://openalex.org/W2168893271","https://openalex.org/W2493096353","https://openalex.org/W2158826266","https://openalex.org/W2108792623","https://openalex.org/W34992575","https://openalex.org/W3147630167"],"abstract_inverted_index":{"Three":[0],"subclasses":[1],"of":[2,12,19,52],"analytic":[3],"and":[4,31],"bi-univalent":[5],"functions":[6,23],"are":[7,16,38],"introduced":[8],"through":[9],"the":[10,41],"use":[11],"q\u2212Gegenbauer":[13],"polynomials,":[14],"which":[15],"a":[17,50],"generalization":[18],"Gegenbauer":[20],"polynomials.":[21],"For":[22],"falling":[24],"within":[25],"these":[26],"subclasses,":[27],"coefficient":[28],"bounds":[29],"a2":[30],"a3":[32],"as":[33,35],"well":[34],"Fekete\u2013Szeg\u00f6":[36],"inequalities":[37],"derived.":[39],"Specializing":[40],"parameters":[42],"used":[43],"in":[44],"our":[45],"main":[46],"results":[47],"leads":[48],"to":[49],"number":[51],"new":[53],"results.":[54]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":14},{"year":2023,"cited_by_count":19}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2023-01-30T00:00:00"}