{"id":"https://openalex.org/W3126212142","doi":"https://doi.org/10.3390/sym13020242","title":"Generalizations of Hardy\u2019s Type Inequalities via Conformable Calculus","display_name":"Generalizations of Hardy\u2019s Type Inequalities via Conformable Calculus","publication_year":2021,"publication_date":"2021-02-01","ids":{"openalex":"https://openalex.org/W3126212142","doi":"https://doi.org/10.3390/sym13020242","mag":"3126212142"},"language":"en","primary_location":{"id":"doi:10.3390/sym13020242","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13020242","pdf_url":"https://www.mdpi.com/2073-8994/13/2/242/pdf?version=1626934182","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/2073-8994/13/2/242/pdf?version=1626934182","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002110465","display_name":"Ghada AlNemer","orcid":"https://orcid.org/0000-0002-2222-7973"},"institutions":[{"id":"https://openalex.org/I106778892","display_name":"Princess Nourah bint Abdulrahman University","ror":"https://ror.org/05b0cyh02","country_code":"SA","type":"education","lineage":["https://openalex.org/I106778892"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Ghada AlNemer","raw_affiliation_strings":["Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-2222-7973","affiliations":[{"raw_affiliation_string":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia","institution_ids":["https://openalex.org/I106778892"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083944943","display_name":"Mohammed Kenawy","orcid":"https://orcid.org/0000-0001-9752-4035"},"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/I66513531","display_name":"Fayoum University","ror":"https://ror.org/023gzwx10","country_code":"EG","type":"education","lineage":["https://openalex.org/I66513531"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Mohammed Kenawy","raw_affiliation_strings":["ASRT, 101 Kasr Al-Aini ST, Academy of Scientific Research & Technology, Cairo 11334, Egypt","Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt"],"raw_orcid":"https://orcid.org/0000-0001-9752-4035","affiliations":[{"raw_affiliation_string":"ASRT, 101 Kasr Al-Aini ST, Academy of Scientific Research & Technology, Cairo 11334, Egypt","institution_ids":["https://openalex.org/I4210118384"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt","institution_ids":["https://openalex.org/I66513531"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5060547119","display_name":"Mohammed Zakarya","orcid":"https://orcid.org/0000-0003-4312-8330"},"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"]},{"id":"https://openalex.org/I82952536","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30","country_code":"SA","type":"education","lineage":["https://openalex.org/I82952536"]}],"countries":["EG","SA"],"is_corresponding":false,"raw_author_name":"Mohammed Zakarya","raw_affiliation_strings":["Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia","Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia","institution_ids":["https://openalex.org/I82952536"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080017749","display_name":"Clemente Cesarano","orcid":"https://orcid.org/0000-0002-1694-7907"},"institutions":[{"id":"https://openalex.org/I4210155399","display_name":"UniNettuno University","ror":"https://ror.org/04q0nep37","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210155399"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Clemente Cesarano","raw_affiliation_strings":["Section of Mathematics, Universit\u00e0 Telematica Internazionale Uninettuno, 00186 Rome, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Section of Mathematics, Universit\u00e0 Telematica Internazionale Uninettuno, 00186 Rome, Italy","institution_ids":["https://openalex.org/I4210155399"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5004188579","display_name":"Haytham M. Rezk","orcid":"https://orcid.org/0000-0001-6782-7908"},"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":"Haytham Rezk","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"],"raw_orcid":"https://orcid.org/0000-0001-6782-7908","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt","institution_ids":["https://openalex.org/I184834183"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5002110465"],"corresponding_institution_ids":["https://openalex.org/I106778892"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":3.9615,"has_fulltext":true,"cited_by_count":19,"citation_normalized_percentile":{"value":0.94267191,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"13","issue":"2","first_page":"242","last_page":"242"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9993000030517578,"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":0.9993000030517578,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9932000041007996,"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":0.9823729991912842},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.7052726745605469},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6612064838409424},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.6542605757713318},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.5666691660881042},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5598741173744202},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4506445825099945},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.348958820104599},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.28498876094818115},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.278054416179657},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07637280225753784}],"concepts":[{"id":"https://openalex.org/C86072612","wikidata":"https://www.wikidata.org/wiki/Q5160239","display_name":"Conformable matrix","level":2,"score":0.9823729991912842},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.7052726745605469},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6612064838409424},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.6542605757713318},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.5666691660881042},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5598741173744202},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4506445825099945},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.348958820104599},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.28498876094818115},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.278054416179657},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07637280225753784},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13020242","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13020242","pdf_url":"https://www.mdpi.com/2073-8994/13/2/242/pdf?version=1626934182","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:15cfc680e1b6484eb87889c6b27f895f","is_oa":true,"landing_page_url":"https://doaj.org/article/15cfc680e1b6484eb87889c6b27f895f","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 13, Iss 2, p 242 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/2/242/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13020242","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/sym13020242","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13020242","pdf_url":"https://www.mdpi.com/2073-8994/13/2/242/pdf?version=1626934182","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":0.5899999737739563,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320321116","display_name":"Academy of Scientific Research and Technology","ror":"https://ror.org/02k284p70"},{"id":"https://openalex.org/F4320322484","display_name":"Princess Nourah Bint Abdulrahman University","ror":"https://ror.org/05b0cyh02"},{"id":"https://openalex.org/F4320337220","display_name":"Deanship of Scientific Research, Princess Nourah Bint Abdulrahman University","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3126212142.pdf","grobid_xml":"https://content.openalex.org/works/W3126212142.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W576547673","https://openalex.org/W1969108223","https://openalex.org/W2052768752","https://openalex.org/W2063067239","https://openalex.org/W2080931732","https://openalex.org/W2088265050","https://openalex.org/W2184737418","https://openalex.org/W2329719292","https://openalex.org/W2330290967","https://openalex.org/W2513831291","https://openalex.org/W2528504294","https://openalex.org/W2609774349","https://openalex.org/W2623671167","https://openalex.org/W2731185211","https://openalex.org/W2751697244","https://openalex.org/W2759652511","https://openalex.org/W2964151816","https://openalex.org/W3009899519","https://openalex.org/W3011005186","https://openalex.org/W6629120302","https://openalex.org/W6702381326","https://openalex.org/W6725850807","https://openalex.org/W6738690619","https://openalex.org/W6740918558","https://openalex.org/W6743682962"],"related_works":["https://openalex.org/W3158217983","https://openalex.org/W4248103533","https://openalex.org/W3046588275","https://openalex.org/W3085259849","https://openalex.org/W2792191619","https://openalex.org/W3046886605","https://openalex.org/W3008631921","https://openalex.org/W4313364931","https://openalex.org/W4302307873","https://openalex.org/W2898762024"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"derive":[4],"some":[5],"new":[6],"fractional":[7,24],"extensions":[8],"of":[9],"Hardy\u2019s":[10],"type":[11],"inequalities.":[12],"The":[13],"corresponding":[14],"reverse":[15],"relations":[16],"are":[17,32],"also":[18],"obtained":[19],"by":[20],"using":[21],"the":[22,28],"conformable":[23],"calculus":[25],"from":[26],"which":[27],"classical":[29],"integral":[30],"inequalities":[31],"deduced":[33],"as":[34],"special":[35],"cases":[36],"at":[37],"\u03b1=1.":[38]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}