{"id":"https://openalex.org/W3198152608","doi":"https://doi.org/10.3390/sym13091605","title":"On Some New Trapezoidal Type Inequalities for Twice (p, q) Differentiable Convex Functions in Post-Quantum Calculus","display_name":"On Some New Trapezoidal Type Inequalities for Twice (p, q) Differentiable Convex Functions in Post-Quantum Calculus","publication_year":2021,"publication_date":"2021-09-01","ids":{"openalex":"https://openalex.org/W3198152608","doi":"https://doi.org/10.3390/sym13091605","mag":"3198152608"},"language":"en","primary_location":{"id":"doi:10.3390/sym13091605","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091605","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1605/pdf?version=1630587460","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/2073-8994/13/9/1605/pdf?version=1630587460","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5065323596","display_name":"Thanin Sitthiwirattham","orcid":"https://orcid.org/0000-0002-8455-1402"},"institutions":[{"id":"https://openalex.org/I4210097440","display_name":"Suan Dusit University","ror":"https://ror.org/00x7vzk16","country_code":"TH","type":"education","lineage":["https://openalex.org/I4210097440"]}],"countries":["TH"],"is_corresponding":true,"raw_author_name":"Thanin Sitthiwirattham","raw_affiliation_strings":["Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"],"raw_orcid":"https://orcid.org/0000-0002-8455-1402","affiliations":[{"raw_affiliation_string":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand","institution_ids":["https://openalex.org/I4210097440"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101407557","display_name":"Ghulam Murtaza","orcid":"https://orcid.org/0000-0002-3615-1922"},"institutions":[{"id":"https://openalex.org/I87482320","display_name":"University of Management and Technology","ror":"https://ror.org/0095xcq10","country_code":"PK","type":"education","lineage":["https://openalex.org/I87482320"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Ghulam Murtaza","raw_affiliation_strings":["Department of Mathematics, University of Management and Technology, Lahore 54700, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Management and Technology, Lahore 54700, Pakistan","institution_ids":["https://openalex.org/I87482320"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5012082265","display_name":"Muhammad Aamir Ali","orcid":"https://orcid.org/0000-0001-5341-4926"},"institutions":[{"id":"https://openalex.org/I152031979","display_name":"Nanjing Normal University","ror":"https://ror.org/036trcv74","country_code":"CN","type":"education","lineage":["https://openalex.org/I152031979"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Muhammad Aamir Ali","raw_affiliation_strings":["Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China"],"raw_orcid":"https://orcid.org/0000-0001-5341-4926","affiliations":[{"raw_affiliation_string":"Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China","institution_ids":["https://openalex.org/I152031979"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086599287","display_name":"Sotiris K. Ntouyas","orcid":"https://orcid.org/0000-0002-7695-2118"},"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"]},{"id":"https://openalex.org/I194019607","display_name":"University of Ioannina","ror":"https://ror.org/01qg3j183","country_code":"GR","type":"education","lineage":["https://openalex.org/I194019607"]}],"countries":["GR","SA"],"is_corresponding":false,"raw_author_name":"Sotiris K. Ntouyas","raw_affiliation_strings":["Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece","Nonlinear Analysis and Applied Mathematics (NAAM)\u2014Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-7695-2118","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece","institution_ids":["https://openalex.org/I194019607"]},{"raw_affiliation_string":"Nonlinear Analysis and Applied Mathematics (NAAM)\u2014Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia","institution_ids":["https://openalex.org/I185163786"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057264345","display_name":"Muhammad Adeel","orcid":"https://orcid.org/0000-0001-5689-3440"},"institutions":[{"id":"https://openalex.org/I87482320","display_name":"University of Management and Technology","ror":"https://ror.org/0095xcq10","country_code":"PK","type":"education","lineage":["https://openalex.org/I87482320"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Adeel","raw_affiliation_strings":["Department of Mathematics, University of Management and Technology, Lahore 54700, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Management and Technology, Lahore 54700, Pakistan","institution_ids":["https://openalex.org/I87482320"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019296026","display_name":"Jarunee Soontharanon","orcid":null},"institutions":[{"id":"https://openalex.org/I82828225","display_name":"King Mongkut's University of Technology North Bangkok","ror":"https://ror.org/04fy6jb97","country_code":"TH","type":"education","lineage":["https://openalex.org/I82828225"]}],"countries":["TH"],"is_corresponding":false,"raw_author_name":"Jarunee Soontharanon","raw_affiliation_strings":["Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand","Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand","institution_ids":["https://openalex.org/I82828225"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand","institution_ids":["https://openalex.org/I82828225"]}]}],"institutions":[],"countries_distinct_count":5,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5012082265","https://openalex.org/A5065323596","https://openalex.org/A5101407557"],"corresponding_institution_ids":["https://openalex.org/I152031979","https://openalex.org/I4210097440","https://openalex.org/I87482320"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.9904,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.73265226,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":97},"biblio":{"volume":"13","issue":"9","first_page":"1605","last_page":"1605"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9423999786376953,"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":0.673197329044342},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.6189889311790466},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.5805148482322693},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5053927898406982},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.47268205881118774},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4483913779258728},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.44258826971054077},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.4126642346382141},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3727395236492157},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34060150384902954},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.32665562629699707},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.24305033683776855},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06491506099700928}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.673197329044342},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.6189889311790466},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.5805148482322693},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5053927898406982},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.47268205881118774},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4483913779258728},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.44258826971054077},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.4126642346382141},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3727395236492157},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34060150384902954},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.32665562629699707},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.24305033683776855},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06491506099700928},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","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/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}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13091605","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091605","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1605/pdf?version=1630587460","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:3136478d2c0640698bef67db8ce0aa45","is_oa":true,"landing_page_url":"https://doaj.org/article/3136478d2c0640698bef67db8ce0aa45","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 9, p 1605 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/9/1605/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13091605","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/sym13091605","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13091605","pdf_url":"https://www.mdpi.com/2073-8994/13/9/1605/pdf?version=1630587460","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":0.5899999737739563}],"awards":[{"id":"https://openalex.org/G8986322138","display_name":null,"funder_award_id":"KMUTNB-63-KNOW-21","funder_id":"https://openalex.org/F4320324344","funder_display_name":"King Mongkut's University of Technology North Bangkok"}],"funders":[{"id":"https://openalex.org/F4320324344","display_name":"King Mongkut's University of Technology North Bangkok","ror":"https://ror.org/04fy6jb97"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3198152608.pdf","grobid_xml":"https://content.openalex.org/works/W3198152608.grobid-xml"},"referenced_works_count":41,"referenced_works":["https://openalex.org/W59400897","https://openalex.org/W61423009","https://openalex.org/W566543047","https://openalex.org/W1094476048","https://openalex.org/W1524266535","https://openalex.org/W1568677510","https://openalex.org/W2054550768","https://openalex.org/W2118017323","https://openalex.org/W2145886492","https://openalex.org/W2148885158","https://openalex.org/W2529427475","https://openalex.org/W2614402218","https://openalex.org/W2790013524","https://openalex.org/W2962387070","https://openalex.org/W2963897276","https://openalex.org/W2977556055","https://openalex.org/W3006443523","https://openalex.org/W3013593946","https://openalex.org/W3019425015","https://openalex.org/W3030308006","https://openalex.org/W3030652065","https://openalex.org/W3045278394","https://openalex.org/W3046153660","https://openalex.org/W3084021619","https://openalex.org/W3101401782","https://openalex.org/W3108036975","https://openalex.org/W3121690530","https://openalex.org/W3124251384","https://openalex.org/W3126612424","https://openalex.org/W3164806255","https://openalex.org/W3164956005","https://openalex.org/W3165108250","https://openalex.org/W3170865853","https://openalex.org/W3174676267","https://openalex.org/W3215201654","https://openalex.org/W4240644230","https://openalex.org/W4300631328","https://openalex.org/W6627175505","https://openalex.org/W6663971105","https://openalex.org/W6721642137","https://openalex.org/W6731907626"],"related_works":["https://openalex.org/W4285277090","https://openalex.org/W4327738859","https://openalex.org/W2348722996","https://openalex.org/W2334570605","https://openalex.org/W3181683615","https://openalex.org/W4286826125","https://openalex.org/W1633485514","https://openalex.org/W2914542058","https://openalex.org/W4385382439","https://openalex.org/W4387635768"],"abstract_inverted_index":{"Quantum":[0],"information":[1,10],"theory,":[2,11],"an":[3],"interdisciplinary":[4],"field":[5,82],"that":[6,68],"includes":[7],"computer":[8],"science,":[9],"philosophy,":[12],"cryptography,":[13],"and":[14,29,46,91],"symmetry,":[15],"has":[16,23],"various":[17,99],"applications":[18,97],"for":[19,59],"quantum":[20],"calculus.":[21],"Inequalities":[22],"a":[24,39,105],"strong":[25],"association":[26],"with":[27],"convex":[28,31,62],"symmetric":[30],"functions.":[32,63],"In":[33],"this":[34,50,89],"study,":[35],"first":[36],"we":[37,48],"establish":[38],"p,q-integral":[40],"identity":[41],"involving":[42],"the":[43,69,74,81,93],"second":[44],"p,q-derivative":[45],"then":[47],"used":[49],"result":[51],"to":[52],"prove":[53],"some":[54,77],"new":[55],"trapezoidal":[56],"type":[57],"inequalities":[58,87],"twice":[60],"p,q-differentiable":[61],"It":[64],"is":[65],"also":[66],"shown":[67],"newly":[70],"established":[71],"results":[72,79],"are":[73],"refinements":[75],"of":[76,83,88],"existing":[78],"in":[80,98,101],"integral":[84],"inequalities.":[85],"Analytic":[86],"nature":[90],"especially":[92],"techniques":[94],"involved":[95],"have":[96],"areas":[100],"which":[102],"symmetry":[103],"plays":[104],"prominent":[106],"role.":[107]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2022,"cited_by_count":3}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2021-09-13T00:00:00"}
