{"id":"https://openalex.org/W3094785456","doi":"https://doi.org/10.1155/2020/8928691","title":"New Estimates for Csisz\u00e1r Divergence and Zipf\u2013Mandelbrot Entropy via Jensen\u2013Mercer\u2019s Inequality","display_name":"New Estimates for Csisz\u00e1r Divergence and Zipf\u2013Mandelbrot Entropy via Jensen\u2013Mercer\u2019s Inequality","publication_year":2020,"publication_date":"2020-11-06","ids":{"openalex":"https://openalex.org/W3094785456","doi":"https://doi.org/10.1155/2020/8928691","mag":"3094785456"},"language":"en","primary_location":{"id":"doi:10.1155/2020/8928691","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2020/8928691","pdf_url":"https://downloads.hindawi.com/journals/complexity/2020/8928691.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/complexity/2020/8928691.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057057760","display_name":"Muhammad Adil Khan","orcid":"https://orcid.org/0000-0001-5373-4663"},"institutions":[{"id":"https://openalex.org/I36001604","display_name":"University of Peshawar","ror":"https://ror.org/02t2qwf81","country_code":"PK","type":"education","lineage":["https://openalex.org/I36001604"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Adil Khan","raw_affiliation_strings":["Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan","institution_ids":["https://openalex.org/I36001604"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103030513","display_name":"Z. Husain","orcid":"https://orcid.org/0000-0002-4826-6065"},"institutions":[{"id":"https://openalex.org/I36001604","display_name":"University of Peshawar","ror":"https://ror.org/02t2qwf81","country_code":"PK","type":"education","lineage":["https://openalex.org/I36001604"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Zakir Husain","raw_affiliation_strings":["Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan","institution_ids":["https://openalex.org/I36001604"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046314241","display_name":"Yu\u2010Ming Chu","orcid":"https://orcid.org/0000-0002-0944-2134"},"institutions":[{"id":"https://openalex.org/I3018263800","display_name":"Huzhou University","ror":"https://ror.org/04mvpxy20","country_code":"CN","type":"education","lineage":["https://openalex.org/I3018263800"]},{"id":"https://openalex.org/I56934997","display_name":"Changsha University of Science and Technology","ror":"https://ror.org/03yph8055","country_code":"CN","type":"education","lineage":["https://openalex.org/I56934997"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yu-Ming Chu","raw_affiliation_strings":["Department of Mathematics, Huzhou University, Huzhou 313000, China","Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Huzhou University, Huzhou 313000, China","institution_ids":["https://openalex.org/I3018263800"]},{"raw_affiliation_string":"Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China","institution_ids":["https://openalex.org/I56934997"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5046314241"],"corresponding_institution_ids":["https://openalex.org/I3018263800","https://openalex.org/I56934997"],"apc_list":{"value":2300,"currency":"USD","value_usd":2300},"apc_paid":{"value":2300,"currency":"USD","value_usd":2300},"fwci":5.848,"has_fulltext":true,"cited_by_count":26,"citation_normalized_percentile":{"value":0.97164711,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"2020","issue":null,"first_page":"1","last_page":"8"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9990000128746033,"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":0.9990000128746033,"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/T10050","display_name":"Multi-Criteria Decision Making","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9933000206947327,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mandelbrot-set","display_name":"Mandelbrot set","score":0.7515119314193726},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.723993182182312},{"id":"https://openalex.org/keywords/jensens-inequality","display_name":"Jensen's inequality","score":0.7109574675559998},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.5986985564231873},{"id":"https://openalex.org/keywords/zipfs-law","display_name":"Zipf's law","score":0.5788151025772095},{"id":"https://openalex.org/keywords/log-sum-inequality","display_name":"Log sum inequality","score":0.5526912808418274},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.5232630968093872},{"id":"https://openalex.org/keywords/rearrangement-inequality","display_name":"Rearrangement inequality","score":0.4797866940498352},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.4599441885948181},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.36938661336898804},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3597036600112915},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.17324298620224},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1528274118900299},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.09591361880302429},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.08530879020690918},{"id":"https://openalex.org/keywords/convex-analysis","display_name":"Convex analysis","score":0.05366763472557068},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.04997771978378296}],"concepts":[{"id":"https://openalex.org/C97519476","wikidata":"https://www.wikidata.org/wiki/Q257","display_name":"Mandelbrot set","level":3,"score":0.7515119314193726},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.723993182182312},{"id":"https://openalex.org/C174169938","wikidata":"https://www.wikidata.org/wiki/Q518131","display_name":"Jensen's inequality","level":5,"score":0.7109574675559998},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.5986985564231873},{"id":"https://openalex.org/C125932096","wikidata":"https://www.wikidata.org/wiki/Q205472","display_name":"Zipf's law","level":2,"score":0.5788151025772095},{"id":"https://openalex.org/C46979015","wikidata":"https://www.wikidata.org/wiki/Q6666859","display_name":"Log sum inequality","level":3,"score":0.5526912808418274},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.5232630968093872},{"id":"https://openalex.org/C126087398","wikidata":"https://www.wikidata.org/wiki/Q1149189","display_name":"Rearrangement inequality","level":4,"score":0.4797866940498352},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.4599441885948181},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36938661336898804},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3597036600112915},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.17324298620224},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1528274118900299},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.09591361880302429},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.08530879020690918},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.05366763472557068},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.04997771978378296},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","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/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1155/2020/8928691","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2020/8928691","pdf_url":"https://downloads.hindawi.com/journals/complexity/2020/8928691.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:hin:complx:8928691","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/8503/2020/8928691.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:2188658ee06d4491ac97cbc3cbbc805f","is_oa":true,"landing_page_url":"https://doaj.org/article/2188658ee06d4491ac97cbc3cbbc805f","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Complexity, Vol 2020 (2020)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2020/8928691","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2020/8928691","pdf_url":"https://downloads.hindawi.com/journals/complexity/2020/8928691.pdf","source":{"id":"https://openalex.org/S207319443","display_name":"Complexity","issn_l":"1076-2787","issn":["1076-2787","1099-0526"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Complexity","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.5199999809265137}],"awards":[{"id":"https://openalex.org/G1064001500","display_name":null,"funder_award_id":"61673169","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G1231421488","display_name":null,"funder_award_id":"under","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G171498429","display_name":null,"funder_award_id":"11601485","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G37568934","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G385937226","display_name":null,"funder_award_id":"11301","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4844902455","display_name":null,"funder_award_id":"11626101","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4858060408","display_name":null,"funder_award_id":"11701176","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5376261061","display_name":null,"funder_award_id":"11301127","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7033253288","display_name":null,"funder_award_id":"Grants","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3094785456.pdf","grobid_xml":"https://content.openalex.org/works/W3094785456.grobid-xml"},"referenced_works_count":65,"referenced_works":["https://openalex.org/W320014142","https://openalex.org/W564239482","https://openalex.org/W656501022","https://openalex.org/W1554844486","https://openalex.org/W1965555277","https://openalex.org/W1970106640","https://openalex.org/W1984393917","https://openalex.org/W1987113198","https://openalex.org/W1989297368","https://openalex.org/W1992143613","https://openalex.org/W1995201166","https://openalex.org/W2019177640","https://openalex.org/W2026938008","https://openalex.org/W2028125409","https://openalex.org/W2033468335","https://openalex.org/W2062948619","https://openalex.org/W2070134780","https://openalex.org/W2077541600","https://openalex.org/W2084157359","https://openalex.org/W2084238990","https://openalex.org/W2086333522","https://openalex.org/W2099111195","https://openalex.org/W2102699036","https://openalex.org/W2122882636","https://openalex.org/W2124792803","https://openalex.org/W2149197198","https://openalex.org/W2150407564","https://openalex.org/W2160943512","https://openalex.org/W2162995444","https://openalex.org/W2163166770","https://openalex.org/W2321470647","https://openalex.org/W2328899746","https://openalex.org/W2419457057","https://openalex.org/W2461213175","https://openalex.org/W2603194489","https://openalex.org/W2742167956","https://openalex.org/W2749979261","https://openalex.org/W2763638377","https://openalex.org/W2884400151","https://openalex.org/W2885704394","https://openalex.org/W2935948266","https://openalex.org/W2948097409","https://openalex.org/W2950627632","https://openalex.org/W2951351560","https://openalex.org/W2967630982","https://openalex.org/W2987973337","https://openalex.org/W2993383518","https://openalex.org/W2995096331","https://openalex.org/W3016527880","https://openalex.org/W3029058286","https://openalex.org/W3031736664","https://openalex.org/W3033013585","https://openalex.org/W3033783331","https://openalex.org/W3038709393","https://openalex.org/W3049490902","https://openalex.org/W3085655513","https://openalex.org/W3096045933","https://openalex.org/W3098144555","https://openalex.org/W3127615759","https://openalex.org/W4238880841","https://openalex.org/W4243672378","https://openalex.org/W4302456205","https://openalex.org/W4372064067","https://openalex.org/W6678323213","https://openalex.org/W6693512474"],"related_works":["https://openalex.org/W58810707","https://openalex.org/W4319791510","https://openalex.org/W1989486236","https://openalex.org/W3033783331","https://openalex.org/W4307051266","https://openalex.org/W1882871713","https://openalex.org/W2317565166","https://openalex.org/W2165910847","https://openalex.org/W3194917076","https://openalex.org/W2101846922"],"abstract_inverted_index":{"Jensen\u2019s":[0,25],"inequality":[1,26],"is":[2,28,38],"one":[3],"of":[4,16,24,35,50,64,77],"the":[5,75],"fundamental":[6],"inequalities":[7],"which":[8,27],"has":[9],"several":[10,56],"applications":[11],"in":[12],"almost":[13],"every":[14],"field":[15],"science.":[17],"In":[18],"2003,":[19],"Mercer":[20],"gave":[21],"a":[22],"variant":[23],"known":[29],"as":[30],"Jensen\u2013Mercer\u2019s":[31,51,78],"inequality.":[32,52,79],"The":[33,62],"purpose":[34],"this":[36,65],"article":[37,66],"to":[39],"propose":[40],"new":[41,57],"bounds":[42,58],"for":[43,59],"Csisz\u00e1r":[44],"and":[45],"related":[46],"divergences":[47],"by":[48],"means":[49],"Also,":[53],"we":[54],"investigate":[55],"Zipf\u2013Mandelbrot":[60],"entropy.":[61],"idea":[63],"may":[67],"further":[68],"stimulate":[69],"research":[70],"on":[71],"information":[72],"theory":[73],"with":[74],"help":[76]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-18T07:56:08.524223","created_date":"2025-10-10T00:00:00"}
