{"id":"https://openalex.org/W4311886060","doi":"https://doi.org/10.3390/sym14122674","title":"Dynamic Inequalities of Two-Dimensional Hardy Type via Alpha-Conformable Derivatives on Time Scales","display_name":"Dynamic Inequalities of Two-Dimensional Hardy Type via Alpha-Conformable Derivatives on Time Scales","publication_year":2022,"publication_date":"2022-12-17","ids":{"openalex":"https://openalex.org/W4311886060","doi":"https://doi.org/10.3390/sym14122674"},"language":"en","primary_location":{"id":"doi:10.3390/sym14122674","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14122674","pdf_url":"https://www.mdpi.com/2073-8994/14/12/2674/pdf?version=1671607935","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/2073-8994/14/12/2674/pdf?version=1671607935","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":false,"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":"middle","author":{"id":"https://openalex.org/A5108729718","display_name":"Alaa A. El\u2010Bary","orcid":null},"institutions":[{"id":"https://openalex.org/I4210118384","display_name":"Academy of Scientific Research and Technology","ror":"https://ror.org/02k284p70","country_code":"EG","type":"nonprofit","lineage":["https://openalex.org/I4210118384"]},{"id":"https://openalex.org/I59272784","display_name":"Arab Academy for Science, Technology, and Maritime Transport","ror":"https://ror.org/0004vyj87","country_code":"EG","type":"education","lineage":["https://openalex.org/I59272784"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Alaa A. El-Bary","raw_affiliation_strings":["Basic and Applied Science Institute, Arab Academy for Science, Technology and Maritime Transport, Alexandria P.O. Box 1029, Egypt","Council of Future Studies and Risk Management, Academy of Scientific Research and Technology, Cairo 11516, Egypt","National Committee for Mathematics, Academy of Scientific Research and Technology, Cairo 11516, Egypt"],"raw_orcid":"https://orcid.org/0000-0002-8846-0487","affiliations":[{"raw_affiliation_string":"Basic and Applied Science Institute, Arab Academy for Science, Technology and Maritime Transport, Alexandria P.O. Box 1029, Egypt","institution_ids":["https://openalex.org/I59272784"]},{"raw_affiliation_string":"Council of Future Studies and Risk Management, Academy of Scientific Research and Technology, Cairo 11516, Egypt","institution_ids":["https://openalex.org/I4210118384"]},{"raw_affiliation_string":"National Committee for Mathematics, Academy of Scientific Research and Technology, Cairo 11516, Egypt","institution_ids":["https://openalex.org/I4210118384"]}]},{"author_position":"middle","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":false,"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"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089496116","display_name":"Kamsing Nonlaopon","orcid":"https://orcid.org/0000-0002-7469-5402"},"institutions":[{"id":"https://openalex.org/I179193067","display_name":"Khon Kaen University","ror":"https://ror.org/03cq4gr50","country_code":"TH","type":"education","lineage":["https://openalex.org/I179193067"]}],"countries":["TH"],"is_corresponding":true,"raw_author_name":"Kamsing Nonlaopon","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"],"raw_orcid":"https://orcid.org/0000-0002-7469-5402","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand","institution_ids":["https://openalex.org/I179193067"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5089496116"],"corresponding_institution_ids":["https://openalex.org/I179193067"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.19538919,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"12","first_page":"2674","last_page":"2674"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9962999820709229,"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":0.9962999820709229,"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/T10514","display_name":"Numerical methods in engineering","score":0.9932000041007996,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9883000254631042,"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/conformable-matrix","display_name":"Conformable matrix","score":0.9645135998725891},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7213747501373291},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.6379085779190063},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.5911833047866821},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4571291506290436},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.35433292388916016},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.337380051612854},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06966757774353027}],"concepts":[{"id":"https://openalex.org/C86072612","wikidata":"https://www.wikidata.org/wiki/Q5160239","display_name":"Conformable matrix","level":2,"score":0.9645135998725891},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7213747501373291},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.6379085779190063},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.5911833047866821},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4571291506290436},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.35433292388916016},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.337380051612854},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06966757774353027},{"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/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/sym14122674","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14122674","pdf_url":"https://www.mdpi.com/2073-8994/14/12/2674/pdf?version=1671607935","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:b261cd0de803452f9b21c30814f467ea","is_oa":true,"landing_page_url":"https://doaj.org/article/b261cd0de803452f9b21c30814f467ea","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 12, p 2674 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/12/2674/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14122674","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/sym14122674","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14122674","pdf_url":"https://www.mdpi.com/2073-8994/14/12/2674/pdf?version=1671607935","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","display_name":"Reduced inequalities","score":0.5699999928474426}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4311886060.pdf","grobid_xml":"https://content.openalex.org/works/W4311886060.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W98230489","https://openalex.org/W237874261","https://openalex.org/W271810198","https://openalex.org/W304963343","https://openalex.org/W325890631","https://openalex.org/W1493505508","https://openalex.org/W1522419168","https://openalex.org/W1922392877","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/W2747812028","https://openalex.org/W3011005186","https://openalex.org/W3046953052","https://openalex.org/W3048014811","https://openalex.org/W3080432495","https://openalex.org/W3092986324","https://openalex.org/W3125633075","https://openalex.org/W3136104196","https://openalex.org/W6682132649"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W4407082414","https://openalex.org/W2937491694","https://openalex.org/W2558746723","https://openalex.org/W4308131477","https://openalex.org/W3037709338","https://openalex.org/W4385473703","https://openalex.org/W4401018364","https://openalex.org/W3009899519","https://openalex.org/W2883154935"],"abstract_inverted_index":{"We":[0],"established":[1],"some":[2,60],"new":[3,11],"\u03b1-conformable":[4,17,37],"dynamic":[5,14,46,90],"inequalities":[6,15,63],"of":[7,13,16,36],"Hardy\u2013Knopp":[8],"type.":[9],"Some":[10],"generalizations":[12],"Hardy":[18],"type":[19],"in":[20,83],"two":[21],"variables":[22],"on":[23,52],"time":[24,53],"scales":[25],"are":[26,41],"established.":[27],"Furthermore,":[28],"we":[29,58,70],"investigated":[30],"Hardy\u2019s":[31,72],"inequality":[32,48,73],"for":[33,74],"several":[34],"functions":[35],"calculus.":[38],"Our":[39],"results":[40],"proved":[42],"by":[43],"using":[44],"two-dimensional":[45],"Jensen\u2019s":[47],"and":[49,76],"Fubini\u2019s":[50],"theorem":[51],"scales.":[54],"When":[55],"\u03b1=1,":[56],"then":[57],"obtain":[59],"well-known":[61],"time-scale":[62],"due":[64],"to":[65,88],"Hardy.":[66],"As":[67],"special":[68],"cases,":[69],"derived":[71],"T=R,T=Z":[75],"T=hZ.":[77],"Symmetry":[78],"plays":[79],"an":[80],"essential":[81],"role":[82],"determining":[84],"the":[85],"correct":[86],"methods":[87],"solve":[89],"inequalities.":[91]},"counts_by_year":[],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}