{"id":"https://openalex.org/W4285094938","doi":"https://doi.org/10.3390/sym14071420","title":"Bounds and Completely Monotonicity of Some Functions Involving the Functions \u03c8\u2032(l) and \u03c8\u2033(l)","display_name":"Bounds and Completely Monotonicity of Some Functions Involving the Functions \u03c8\u2032(l) and \u03c8\u2033(l)","publication_year":2022,"publication_date":"2022-07-11","ids":{"openalex":"https://openalex.org/W4285094938","doi":"https://doi.org/10.3390/sym14071420"},"language":"en","primary_location":{"id":"doi:10.3390/sym14071420","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14071420","pdf_url":"https://www.mdpi.com/2073-8994/14/7/1420/pdf?version=1657526658","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/2073-8994/14/7/1420/pdf?version=1657526658","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/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"]}]},{"author_position":"last","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"]}]}],"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":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.47847864,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"14","issue":"7","first_page":"1420","last_page":"1420"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":1.0,"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":1.0,"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":0.9889000058174133,"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/T11822","display_name":"Analytic and geometric function theory","score":0.9754999876022339,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.907451868057251},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6865617632865906},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.5942601561546326},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5782537460327148},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5571594834327698},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.507777750492096},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.4851880669593811},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.4636325538158417},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.426454097032547},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32700562477111816},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2324140965938568},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07876977324485779},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.045978665351867676}],"concepts":[{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.907451868057251},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6865617632865906},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.5942601561546326},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5782537460327148},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5571594834327698},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.507777750492096},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.4851880669593811},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4636325538158417},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.426454097032547},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32700562477111816},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2324140965938568},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07876977324485779},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.045978665351867676},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14071420","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14071420","pdf_url":"https://www.mdpi.com/2073-8994/14/7/1420/pdf?version=1657526658","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:77316cbfe16a48129fd6d9dc483c3e24","is_oa":true,"landing_page_url":"https://doaj.org/article/77316cbfe16a48129fd6d9dc483c3e24","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 7, p 1420 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/7/1420/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14071420","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/sym14071420","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14071420","pdf_url":"https://www.mdpi.com/2073-8994/14/7/1420/pdf?version=1657526658","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":[{"id":"https://metadata.un.org/sdg/10","score":0.4399999976158142,"display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4285094938.pdf","grobid_xml":"https://content.openalex.org/works/W4285094938.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W1560063022","https://openalex.org/W1670149396","https://openalex.org/W1710013190","https://openalex.org/W1963912938","https://openalex.org/W1993522052","https://openalex.org/W1997299684","https://openalex.org/W2000950501","https://openalex.org/W2008891576","https://openalex.org/W2015649080","https://openalex.org/W2027191563","https://openalex.org/W2033092527","https://openalex.org/W2041654271","https://openalex.org/W2051449379","https://openalex.org/W2076748548","https://openalex.org/W2117885262","https://openalex.org/W2145249146","https://openalex.org/W2747919117","https://openalex.org/W2919382251","https://openalex.org/W3098325605","https://openalex.org/W3099667084","https://openalex.org/W3105982852","https://openalex.org/W3167547514","https://openalex.org/W3185902655","https://openalex.org/W3208878862","https://openalex.org/W4207069325","https://openalex.org/W4300694852","https://openalex.org/W6637575395","https://openalex.org/W6681858654","https://openalex.org/W6786127680","https://openalex.org/W6795887699","https://openalex.org/W6807837689"],"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/W2945307361","https://openalex.org/W1987764909","https://openalex.org/W2116636209","https://openalex.org/W2053698688"],"abstract_inverted_index":{"Symmetrical":[0],"patterns":[1],"exist":[2],"in":[3,13],"the":[4,33,37,46,62,69,80],"nature":[5],"of":[6,25,42,65,79],"inequalities,":[7],"which":[8],"play":[9],"a":[10,53],"basic":[11],"role":[12],"theoretical":[14],"and":[15,49],"applied":[16],"mathematics.":[17],"In":[18,32],"several":[19],"studies,":[20],"inequalities":[21],"present":[22],"accurate":[23],"approximations":[24],"functions":[26,44,67],"based":[27],"on":[28],"their":[29],"symmetry":[30],"properties.":[31],"paper,":[34],"we":[35,51,60],"prove":[36],"completely":[38],"monotonic":[39],"(CM)":[40],"property":[41],"some":[43,66,78],"involving":[45,68],"function":[47,70],"\u0394(l)=\u03c8\u2033(l)+\u03c8\u2032(l)2":[48],"hence":[50],"deduce":[52],"new":[54,73],"double":[55],"inequality":[56],"for":[57],"it.":[58],"Additionally,":[59],"study":[61],"CM":[63],"degree":[64],"\u03c8\u2032(l).":[71],"Our":[72],"bounds":[74],"takes":[75],"priority":[76],"over":[77],"recently":[81],"published":[82],"results.":[83]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2022-07-14T00:00:00"}