{"id":"https://openalex.org/W4304184183","doi":"https://doi.org/10.3390/sym14102047","title":"Diamond-\u03b1 Hardy-Type Inequalities on Time Scales","display_name":"Diamond-\u03b1 Hardy-Type Inequalities on Time Scales","publication_year":2022,"publication_date":"2022-09-30","ids":{"openalex":"https://openalex.org/W4304184183","doi":"https://doi.org/10.3390/sym14102047"},"language":"en","primary_location":{"id":"doi:10.3390/sym14102047","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14102047","pdf_url":"https://www.mdpi.com/2073-8994/14/10/2047/pdf?version=1666663771","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/2073-8994/14/10/2047/pdf?version=1666663771","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000462219","display_name":"Ahmed A. El\u2010Deeb","orcid":null},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]}],"countries":["EG"],"is_corresponding":true,"raw_author_name":"Ahmed A. El-Deeb","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt"],"raw_orcid":"https://orcid.org/0000-0003-2822-4092","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025074757","display_name":"Jan Awrejcewicz","orcid":"https://orcid.org/0000-0003-0387-921X"},"institutions":[{"id":"https://openalex.org/I188884621","display_name":"Lodz University of Technology","ror":"https://ror.org/00s8fpf52","country_code":"PL","type":"education","lineage":["https://openalex.org/I188884621"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Jan Awrejcewicz","raw_affiliation_strings":["Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland"],"raw_orcid":"https://orcid.org/0000-0003-0387-921X","affiliations":[{"raw_affiliation_string":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland","institution_ids":["https://openalex.org/I188884621"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5000462219","https://openalex.org/A5025074757"],"corresponding_institution_ids":["https://openalex.org/I184834183","https://openalex.org/I188884621"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.7792,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.8417987,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"14","issue":"10","first_page":"2047","last_page":"2047"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9976999759674072,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9976999759674072,"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":0.9787999987602234,"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9753999710083008,"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/mathematics","display_name":"Mathematics","score":0.7440266013145447},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.6274067163467407},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.6053104400634766},{"id":"https://openalex.org/keywords/fubinis-theorem","display_name":"Fubini's theorem","score":0.5274118185043335},{"id":"https://openalex.org/keywords/dynamic-equation","display_name":"Dynamic equation","score":0.5067816376686096},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.4982576370239258},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.41903257369995117},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.37984421849250793},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3265559673309326},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08532920479774475},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06026095151901245}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7440266013145447},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.6274067163467407},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.6053104400634766},{"id":"https://openalex.org/C55526981","wikidata":"https://www.wikidata.org/wiki/Q1149022","display_name":"Fubini's theorem","level":2,"score":0.5274118185043335},{"id":"https://openalex.org/C2989370511","wikidata":"https://www.wikidata.org/wiki/Q18381590","display_name":"Dynamic equation","level":3,"score":0.5067816376686096},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.4982576370239258},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.41903257369995117},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37984421849250793},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3265559673309326},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08532920479774475},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06026095151901245},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14102047","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14102047","pdf_url":"https://www.mdpi.com/2073-8994/14/10/2047/pdf?version=1666663771","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:99f0a0af855b4453907b3a3765129fbd","is_oa":true,"landing_page_url":"https://doaj.org/article/99f0a0af855b4453907b3a3765129fbd","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 10, p 2047 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/10/2047/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14102047","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/sym14102047","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14102047","pdf_url":"https://www.mdpi.com/2073-8994/14/10/2047/pdf?version=1666663771","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":[{"score":0.5400000214576721,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4304184183.pdf","grobid_xml":"https://content.openalex.org/works/W4304184183.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W98230489","https://openalex.org/W271810198","https://openalex.org/W325890631","https://openalex.org/W1493505508","https://openalex.org/W1522419168","https://openalex.org/W1922392877","https://openalex.org/W2015450767","https://openalex.org/W2047965186","https://openalex.org/W2066397800","https://openalex.org/W2069528055","https://openalex.org/W2075747575","https://openalex.org/W2145797734","https://openalex.org/W2171064145","https://openalex.org/W2800962038","https://openalex.org/W2885350278","https://openalex.org/W2909207809","https://openalex.org/W2943113586","https://openalex.org/W2967156670","https://openalex.org/W3165765870","https://openalex.org/W6682132649","https://openalex.org/W6685321362","https://openalex.org/W6753130120"],"related_works":["https://openalex.org/W2914103171","https://openalex.org/W4220715110","https://openalex.org/W1503132626","https://openalex.org/W3166318962","https://openalex.org/W2580597849","https://openalex.org/W2046196304","https://openalex.org/W2289677564","https://openalex.org/W4308439067","https://openalex.org/W3213074928","https://openalex.org/W4299682115"],"abstract_inverted_index":{"In":[0],"the":[1,48,61,66,79,92],"present":[2,40],"article,":[3],"we":[4,39],"prove":[5],"some":[6],"new":[7,23],"generalizations":[8,24],"of":[9,12,25,28,78],"dynamic":[10,17,26,50,62,97],"inequalities":[11,27,42],"Hardy-type":[13,29,83],"by":[14,46,59],"utilizing":[15],"diamond-\u03b1":[16,49],"integrals":[18,51],"on":[19,33,52,69],"time":[20,34,53,70],"scales.":[21,54,71],"Furthermore,":[22],"in":[30,90],"two":[31],"variables":[32],"scales":[35],"are":[36,57],"proved.":[37],"Moreover,":[38],"Hardy":[41],"for":[43],"several":[44],"functions":[45],"using":[47,60],"The":[55],"results":[56,74,77],"proved":[58],"Jensen":[63],"inequality":[64],"and":[65,81],"Fubini":[67],"theorem":[68],"Our":[72],"main":[73],"extend":[75],"existing":[76],"integral":[80],"discrete":[82],"inequalities.":[84,98],"Symmetry":[85],"plays":[86],"an":[87],"essential":[88],"role":[89],"determining":[91],"correct":[93],"methods":[94],"to":[95],"solve":[96]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":3}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2022-10-11T00:00:00"}