{"id":"https://openalex.org/W4225389435","doi":"https://doi.org/10.3390/sym14050929","title":"Some Rational Approximations and Bounds for Bateman\u2019s G-Function","display_name":"Some Rational Approximations and Bounds for Bateman\u2019s G-Function","publication_year":2022,"publication_date":"2022-05-02","ids":{"openalex":"https://openalex.org/W4225389435","doi":"https://doi.org/10.3390/sym14050929"},"language":"en","primary_location":{"id":"doi:10.3390/sym14050929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050929","pdf_url":"https://www.mdpi.com/2073-8994/14/5/929/pdf?version=1651487051","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/5/929/pdf?version=1651487051","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017556216","display_name":"Omelsaad Ahfaf","orcid":null},"institutions":[{"id":"https://openalex.org/I159247623","display_name":"Mansoura University","ror":"https://ror.org/01k8vtd75","country_code":"EG","type":"education","lineage":["https://openalex.org/I159247623"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Omelsaad Ahfaf","raw_affiliation_strings":["Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt","institution_ids":["https://openalex.org/I159247623"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024058768","display_name":"Mansour Mahmoud","orcid":"https://orcid.org/0000-0002-5918-1913"},"institutions":[{"id":"https://openalex.org/I185163786","display_name":"King Abdulaziz University","ror":"https://ror.org/02ma4wv74","country_code":"SA","type":"education","lineage":["https://openalex.org/I185163786"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Mansour Mahmoud","raw_affiliation_strings":["Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-5918-1913","affiliations":[{"raw_affiliation_string":"Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia","institution_ids":["https://openalex.org/I185163786"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019523560","display_name":"Ahmed Talat","orcid":"https://orcid.org/0000-0001-7702-8093"},"institutions":[{"id":"https://openalex.org/I88017793","display_name":"Port Said University","ror":"https://ror.org/01vx5yq44","country_code":"EG","type":"education","lineage":["https://openalex.org/I88017793"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Ahmed Talat","raw_affiliation_strings":["Mathematics and Computer Sciences Department, Faculty of Science, Port Said University, Port Said 42526, Egypt"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics and Computer Sciences Department, Faculty of Science, Port Said University, Port Said 42526, Egypt","institution_ids":["https://openalex.org/I88017793"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5024058768"],"corresponding_institution_ids":["https://openalex.org/I185163786"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.3556,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.45616321,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"14","issue":"5","first_page":"929","last_page":"929"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9997000098228455,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9961000084877014,"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/T12508","display_name":"Functional Equations Stability Results","score":0.9958999752998352,"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/monotonic-function","display_name":"Monotonic function","score":0.7893580198287964},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6185763478279114},{"id":"https://openalex.org/keywords/rational-function","display_name":"Rational function","score":0.5805097222328186},{"id":"https://openalex.org/keywords/approximations-of-\u03c0","display_name":"Approximations of \u03c0","score":0.5796673893928528},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5694152116775513},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.40986305475234985},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.40098804235458374},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23573681712150574}],"concepts":[{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.7893580198287964},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6185763478279114},{"id":"https://openalex.org/C75190567","wikidata":"https://www.wikidata.org/wiki/Q41237","display_name":"Rational function","level":2,"score":0.5805097222328186},{"id":"https://openalex.org/C193386753","wikidata":"https://www.wikidata.org/wiki/Q1130396","display_name":"Approximations of \u03c0","level":2,"score":0.5796673893928528},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5694152116775513},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.40986305475234985},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40098804235458374},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23573681712150574},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14050929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050929","pdf_url":"https://www.mdpi.com/2073-8994/14/5/929/pdf?version=1651487051","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:e28634af05fb4b43b6f79ee5bedf8b90","is_oa":true,"landing_page_url":"https://doaj.org/article/e28634af05fb4b43b6f79ee5bedf8b90","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":"Symmetry, Vol 14, Iss 5, p 929 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/5/929/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14050929","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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14050929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050929","pdf_url":"https://www.mdpi.com/2073-8994/14/5/929/pdf?version=1651487051","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4225389435.pdf"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1584758702","https://openalex.org/W2000950501","https://openalex.org/W2071610623","https://openalex.org/W2071789471","https://openalex.org/W2090100933","https://openalex.org/W2477395411","https://openalex.org/W2528181886","https://openalex.org/W2734472729","https://openalex.org/W3107556262","https://openalex.org/W3118959635","https://openalex.org/W6721382596","https://openalex.org/W6740794858"],"related_works":["https://openalex.org/W2077314575","https://openalex.org/W4315701745","https://openalex.org/W1990290471","https://openalex.org/W4380682190","https://openalex.org/W2005710836","https://openalex.org/W2102386043","https://openalex.org/W4321392595","https://openalex.org/W120419281","https://openalex.org/W2094480099","https://openalex.org/W2063893845"],"abstract_inverted_index":{"Symmetrical":[0],"patterns":[1],"exist":[2],"in":[3,13],"the":[4,37],"nature":[5],"of":[6,25,57],"inequalities,":[7],"which":[8],"play":[9],"a":[10,50,60],"basic":[11],"role":[12],"theoretical":[14],"and":[15,47,59],"applied":[16],"mathematics.":[17],"In":[18,32],"several":[19],"studies,":[20],"inequalities":[21],"present":[22,36],"accurate":[23],"approximations":[24,40],"functions":[26],"based":[27],"on":[28],"their":[29],"symmetry":[30],"properties.":[31],"this":[33],"paper,":[34],"we":[35,52],"following":[38],"rational":[39],"for":[41],"Bateman\u2019s":[42],"G-function":[43],"G(w)=1w+2w2+\u2211j=1n4\u03b1jw2\u22122j\u22121+O1w2n+2,":[44],"where":[45],"\u03b11=14,":[46],"\u03b1j=(1\u221222j+2)B2j+2j+1+\u2211\u03bd=1j\u22121(1\u221222j\u22122\u03bd+2)B2j\u22122\u03bd+2\u03b1\u03bdj\u2212\u03bd+1,j>1.":[48],"As":[49],"consequence,":[51],"introduced":[53],"some":[54],"new":[55],"bounds":[56],"G(w)":[58],"completely":[61],"monotonic":[62],"function":[63],"involving":[64],"it.":[65]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-22T06:13:13.366637","created_date":"2022-05-05T00:00:00"}