{"id":"https://openalex.org/W7212244187","doi":"https://doi.org/10.48550/arxiv.2609.11556","title":"A counterexample to an open problem of Dorff","display_name":"A counterexample to an open problem of Dorff","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7212244187","doi":"https://doi.org/10.48550/arxiv.2609.11556"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.11556","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11556","pdf_url":"https://arxiv.org/pdf/2609.11556","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.11556","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5122176963","display_name":"Zhi-Gang Wang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Zhi-Gang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5005792239","display_name":"Deguang Zhong","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhong, Deguang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11822","display_name":"Analytic and geometric function theory","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11822","display_name":"Analytic and geometric function theory","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T10884","display_name":"Holomorphic and Operator Theory","score":9.999999747378752e-05,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/planar-harmonic-mappings","display_name":"planar harmonic mappings","score":0.8999999761581421},{"id":"https://openalex.org/keywords/self-convolution","display_name":"self-convolution","score":0.9200000166893005},{"id":"https://openalex.org/keywords/convex-univalent-functions","display_name":"convex univalent functions","score":0.7770000100135803},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.9700000286102295}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8177000284194946},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.6610000133514404},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6093999743461609},{"id":"https://openalex.org/C47177299","wikidata":"https://www.wikidata.org/wiki/Q2390323","display_name":"Unit disk","level":2,"score":0.590499997138977},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.5320000052452087},{"id":"https://openalex.org/C2776299755","wikidata":"https://www.wikidata.org/wiki/Q432449","display_name":"Carry (investment)","level":2,"score":0.47519999742507935},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.4708000123500824},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.453000009059906},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.4300000071525574},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.4255000054836273},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.4244000017642975},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.4025999903678894},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38449999690055847},{"id":"https://openalex.org/C150397156","wikidata":"https://www.wikidata.org/wiki/Q591310","display_name":"General position","level":2,"score":0.3779999911785126},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3776000142097473},{"id":"https://openalex.org/C45347329","wikidata":"https://www.wikidata.org/wiki/Q5166604","display_name":"Convolution (computer science)","level":3,"score":0.3700999915599823},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.36629998683929443},{"id":"https://openalex.org/C627467","wikidata":"https://www.wikidata.org/wiki/Q599027","display_name":"Harmonic function","level":2,"score":0.3522000014781952},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.3366999924182892},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.3366999924182892},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.31709998846054077},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.3005000054836273},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.2978000044822693},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.2874999940395355},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2854999899864197},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.28049999475479126},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.2784000039100647},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.2676999866962433},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.2655999958515167},{"id":"https://openalex.org/C70251129","wikidata":"https://www.wikidata.org/wiki/Q260928","display_name":"Jordan curve theorem","level":2,"score":0.26109999418258667},{"id":"https://openalex.org/C53688716","wikidata":"https://www.wikidata.org/wiki/Q5659270","display_name":"Harmonic measure","level":3,"score":0.2583000063896179},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.2547999918460846}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.11556","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11556","pdf_url":"https://arxiv.org/pdf/2609.11556","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},{"id":"doi:10.48550/arxiv.2609.11556","doi":"https://doi.org/10.48550/arxiv.2609.11556","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.11556","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.11556","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11556","pdf_url":"https://arxiv.org/pdf/2609.11556","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G4343562384","display_name":null,"funder_award_id":"2023A1515011809","funder_id":"https://openalex.org/F4320337111","funder_display_name":"Basic and Applied Basic Research Foundation of Guangdong Province"},{"id":"https://openalex.org/G553955554","display_name":null,"funder_award_id":"2022A1515110967","funder_id":"https://openalex.org/F4320337111","funder_display_name":"Basic and Applied Basic Research Foundation of Guangdong Province"},{"id":"https://openalex.org/G5632167411","display_name":null,"funder_award_id":"25A0668","funder_id":"https://openalex.org/F4320311213","funder_display_name":"Education Department of Hunan Province"}],"funders":[{"id":"https://openalex.org/F4320311213","display_name":"Education Department of Hunan Province","ror":"https://ror.org/05ckg3w11"},{"id":"https://openalex.org/F4320337111","display_name":"Basic and Applied Basic Research Foundation of Guangdong Province","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212244187.pdf","grobid_xml":"https://content.openalex.org/works/W7212244187.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2112158761","https://openalex.org/W2910241894","https://openalex.org/W2952306345","https://openalex.org/W1973066830","https://openalex.org/W2133934352","https://openalex.org/W2799153605","https://openalex.org/W2019828171","https://openalex.org/W1980325099","https://openalex.org/W2964321713","https://openalex.org/W2736039619"],"abstract_inverted_index":{"The":[0],"classical":[1],"P\u00f3lya-Schoenberg":[2],"conjecture,":[3],"proved":[4],"by":[5],"Ruscheweyh-Sheil-Small,":[6],"asserts":[7],"that":[8,62],"the":[9,34,38,52,64,80],"convolution":[10],"of":[11,40,79],"two":[12],"normalized":[13,42,58],"convex":[14,43],"univalent":[15],"functions":[16],"is":[17],"again":[18],"convex.":[19],"This":[20],"property":[21],"fails":[22],"to":[23,26,88],"carry":[24],"over":[25],"planar":[27],"harmonic":[28,44,60],"mappings.":[29],"In":[30],"2001,":[31],"Dorff":[32],"posed":[33],"open":[35,90],"problem":[36],"whether":[37],"self-convolution":[39,71],"a":[41,57,85],"mapping":[45],"with":[46],"bounded":[47],"image":[48],"must":[49],"remain":[50],"in":[51],"same":[53],"class.":[54],"We":[55],"construct":[56],"sense-preserving":[59],"diffeomorphism":[61],"maps":[63],"unit":[65,81],"disk":[66],"onto":[67],"an":[68],"ellipse;":[69],"its":[70],"has":[72],"vanishing":[73],"Jacobian":[74],"at":[75],"some":[76],"interior":[77],"point":[78],"disk,":[82],"which":[83],"provides":[84],"negative":[86],"answer":[87],"Dorff's":[89],"problem.":[91]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-12T00:00:00"}
