{"id":"https://openalex.org/W4210304219","doi":"https://doi.org/10.3390/sym14020302","title":"On Generalization of Different Integral Inequalities for Harmonically Convex Functions","display_name":"On Generalization of Different Integral Inequalities for Harmonically Convex Functions","publication_year":2022,"publication_date":"2022-02-02","ids":{"openalex":"https://openalex.org/W4210304219","doi":"https://doi.org/10.3390/sym14020302"},"language":"en","primary_location":{"id":"doi:10.3390/sym14020302","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14020302","pdf_url":"https://www.mdpi.com/2073-8994/14/2/302/pdf?version=1644887807","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/2073-8994/14/2/302/pdf?version=1644887807","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024985876","display_name":"Jiraporn Reunsumrit","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":"Jiraporn Reunsumrit","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"],"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"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066464647","display_name":"Miguel Vivas\u2013Cortez","orcid":"https://orcid.org/0000-0002-1567-0264"},"institutions":[{"id":"https://openalex.org/I39398033","display_name":"Pontificia Universidad Cat\u00f3lica del Ecuador","ror":"https://ror.org/02qztda51","country_code":"EC","type":"education","lineage":["https://openalex.org/I39398033"]}],"countries":["EC"],"is_corresponding":true,"raw_author_name":"Miguel J. Vivas-Cortez","raw_affiliation_strings":["Escuela de Ciencias Matem\u00e1ticas y F\u00edsicas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Cat\u00f3lica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, Ecuador"],"affiliations":[{"raw_affiliation_string":"Escuela de Ciencias Matem\u00e1ticas y F\u00edsicas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Cat\u00f3lica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, Ecuador","institution_ids":["https://openalex.org/I39398033"]}]},{"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":false,"raw_author_name":"Muhammad Aamir Ali","raw_affiliation_strings":["Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China"],"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":"last","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":false,"raw_author_name":"Thanin Sitthiwirattham","raw_affiliation_strings":["Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"],"affiliations":[{"raw_affiliation_string":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand","institution_ids":["https://openalex.org/I4210097440"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5066464647"],"corresponding_institution_ids":["https://openalex.org/I39398033"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.7117,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.60511247,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"14","issue":"2","first_page":"302","last_page":"302"},"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/T12508","display_name":"Functional Equations Stability Results","score":0.9713000059127808,"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":0.9603999853134155,"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/differentiable-function","display_name":"Differentiable function","score":0.7615272402763367},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.752622127532959},{"id":"https://openalex.org/keywords/midpoint","display_name":"Midpoint","score":0.6903770565986633},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.6850653886795044},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.6520001888275146},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.6350060701370239},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.5707512497901917},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5367907285690308},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4801899194717407},{"id":"https://openalex.org/keywords/variety","display_name":"Variety (cybernetics)","score":0.48009994626045227},{"id":"https://openalex.org/keywords/identity","display_name":"Identity (music)","score":0.4287254810333252},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.42510685324668884},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3689149022102356},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3444419503211975},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3291623592376709},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.16614574193954468},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07597464323043823}],"concepts":[{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.7615272402763367},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.752622127532959},{"id":"https://openalex.org/C34130140","wikidata":"https://www.wikidata.org/wiki/Q1645406","display_name":"Midpoint","level":2,"score":0.6903770565986633},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.6850653886795044},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.6520001888275146},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.6350060701370239},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.5707512497901917},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5367907285690308},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4801899194717407},{"id":"https://openalex.org/C136197465","wikidata":"https://www.wikidata.org/wiki/Q1729295","display_name":"Variety (cybernetics)","level":2,"score":0.48009994626045227},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.4287254810333252},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.42510685324668884},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3689149022102356},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3444419503211975},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3291623592376709},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.16614574193954468},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07597464323043823},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14020302","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14020302","pdf_url":"https://www.mdpi.com/2073-8994/14/2/302/pdf?version=1644887807","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:4f9f454ff0424ff78e10be9dba606c18","is_oa":true,"landing_page_url":"https://doaj.org/article/4f9f454ff0424ff78e10be9dba606c18","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 14, Iss 2, p 302 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/2/302/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14020302","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/sym14020302","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14020302","pdf_url":"https://www.mdpi.com/2073-8994/14/2/302/pdf?version=1644887807","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":0.6600000262260437}],"awards":[{"id":"https://openalex.org/G4535768570","display_name":null,"funder_award_id":"KMUTNB-62-KNOW-26","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/W4210304219.pdf","grobid_xml":"https://content.openalex.org/works/W4210304219.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W816848581","https://openalex.org/W2006458314","https://openalex.org/W2056135743","https://openalex.org/W2768852698","https://openalex.org/W2803296305","https://openalex.org/W2949130693","https://openalex.org/W2999316265","https://openalex.org/W3030369583","https://openalex.org/W3032034118","https://openalex.org/W3048865070","https://openalex.org/W3112331127","https://openalex.org/W3132844534","https://openalex.org/W3156639000","https://openalex.org/W3157211432","https://openalex.org/W3171223766","https://openalex.org/W3200350648","https://openalex.org/W3207870173","https://openalex.org/W3211008961","https://openalex.org/W6623047232","https://openalex.org/W6778489381","https://openalex.org/W6791606018","https://openalex.org/W6794154271","https://openalex.org/W6794365128","https://openalex.org/W6803277159"],"related_works":["https://openalex.org/W4285988348","https://openalex.org/W4207006390","https://openalex.org/W1991171386","https://openalex.org/W1969361911","https://openalex.org/W4298646752","https://openalex.org/W2387310429","https://openalex.org/W2914542058","https://openalex.org/W4385382439","https://openalex.org/W4239678022","https://openalex.org/W4224011354"],"abstract_inverted_index":{"In":[0],"this":[1,21,40],"study,":[2],"we":[3,19],"first":[4],"prove":[5],"a":[6,29,53],"parameterized":[7],"integral":[8],"identity":[9],"involving":[10],"differentiable":[11,15],"functions.":[12],"Then,":[13],"for":[14,47],"harmonically":[16],"convex":[17],"functions,":[18],"use":[20],"result":[22],"to":[23,75],"establish":[24],"some":[25],"new":[26],"inequalities":[27,38],"of":[28,39,55,65,79],"midpoint":[30],"type,":[31,33,41],"trapezoidal":[32],"and":[34],"Simpson":[35],"type.":[36],"Analytic":[37],"as":[42,44,71,73],"well":[43,72],"the":[45,76],"approaches":[46],"solving":[48],"them,":[49],"have":[50],"applications":[51,74],"in":[52],"variety":[54],"domains":[56],"where":[57],"symmetry":[58],"is":[59],"important.":[60],"Finally,":[61],"several":[62],"particular":[63],"cases":[64],"recently":[66],"discovered":[67],"results":[68],"are":[69],"discussed,":[70],"special":[77],"means":[78],"real":[80],"numbers.":[81]},"counts_by_year":[{"year":2022,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}