{"id":"https://openalex.org/W7214230976","doi":"https://doi.org/10.48550/arxiv.2609.26850","title":"A bootstrap proof of Bohr's theorem","display_name":"A bootstrap proof of Bohr's theorem","publication_year":2026,"publication_date":"2026-09-22","ids":{"openalex":"https://openalex.org/W7214230976","doi":"https://doi.org/10.48550/arxiv.2609.26850"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.26850","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26850","pdf_url":"https://arxiv.org/pdf/2609.26850","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/2609.26850","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110784598","display_name":"Ole Fredrik Brevig","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I204778367","display_name":"Norwegian University of Science and Technology","ror":"https://ror.org/05xg72x27","country_code":"NO","type":"education","lineage":["https://openalex.org/I204778367"]}],"countries":["NO"],"is_corresponding":true,"raw_author_name":"Brevig, Ole Fredrik","raw_affiliation_strings":["Department of Mathematical Sciences , Norwegian University of Science and Technology (NTNU) , 7491 Trondheim , Norway"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences , Norwegian University of Science and Technology (NTNU) , 7491 Trondheim , Norway","institution_ids":["https://openalex.org/I204778367"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5110784598"],"corresponding_institution_ids":["https://openalex.org/I204778367"],"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":0.8779000043869019,"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":0.8779000043869019,"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":0.08429999649524689,"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/T12812","display_name":"Meromorphic and Entire Functions","score":0.019700000062584877,"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/majorant-functions","display_name":"majorant functions","score":0.7039999961853027},{"id":"https://openalex.org/keywords/analytic-functions-in-the-unit-disk","display_name":"analytic functions in the unit disk","score":0.9700000286102295},{"id":"https://openalex.org/keywords/bootstrap-argument","display_name":"bootstrap argument","score":0.42500001192092896},{"id":"https://openalex.org/keywords/sharp-coefficient-bounds","display_name":"sharp coefficient bounds","score":0.5809999704360962},{"id":"https://openalex.org/keywords/univalent-functions","display_name":"univalent functions","score":0.46299999952316284}],"concepts":[{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7975999712944031},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7718999981880188},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.565500020980835},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5051000118255615},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.47130000591278076},{"id":"https://openalex.org/C3098717","wikidata":"https://www.wikidata.org/wiki/Q1144872","display_name":"Modulus of continuity","level":3,"score":0.4081999957561493},{"id":"https://openalex.org/C205979905","wikidata":"https://www.wikidata.org/wiki/Q215084","display_name":"Analytic function","level":2,"score":0.4058000147342682},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.396699994802475},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3797999918460846},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.35740000009536743},{"id":"https://openalex.org/C98184364","wikidata":"https://www.wikidata.org/wiki/Q1780131","display_name":"Argument (complex analysis)","level":2,"score":0.35010001063346863},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.34619998931884766},{"id":"https://openalex.org/C27851653","wikidata":"https://www.wikidata.org/wiki/Q829653","display_name":"Bounded variation","level":2,"score":0.3319000005722046},{"id":"https://openalex.org/C92928283","wikidata":"https://www.wikidata.org/wiki/Q4949981","display_name":"Bounded deformation","level":4,"score":0.32420000433921814},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.3009999990463257},{"id":"https://openalex.org/C161505775","wikidata":"https://www.wikidata.org/wiki/Q3240280","display_name":"Direct proof","level":2,"score":0.2989000082015991},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.26910001039505005},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.26739999651908875},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26179999113082886},{"id":"https://openalex.org/C43994743","wikidata":"https://www.wikidata.org/wiki/Q516776","display_name":"Measurable function","level":3,"score":0.26170000433921814}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.26850","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26850","pdf_url":"https://arxiv.org/pdf/2609.26850","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.2609.26850","doi":"https://doi.org/10.48550/arxiv.2609.26850","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.26850","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.26850","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26850","pdf_url":"https://arxiv.org/pdf/2609.26850","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":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214230976.pdf","grobid_xml":"https://content.openalex.org/works/W7214230976.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7147521262","https://openalex.org/W3175984134","https://openalex.org/W3200725179","https://openalex.org/W4414449139","https://openalex.org/W4400421826","https://openalex.org/W3036270441","https://openalex.org/W3016962874","https://openalex.org/W4389720088","https://openalex.org/W4321790562","https://openalex.org/W3084995503"],"abstract_inverted_index":{"Let":[0],"$f$":[1],"be":[2],"an":[3],"analytic":[4],"function":[5,29],"in":[6,55],"the":[7],"unit":[8],"disc":[9],"whose":[10],"modulus":[11],"is":[12,31,43],"bounded":[13,32],"by":[14,33],"$1$.":[15],"We":[16],"present":[17],"a":[18,49,62],"new":[19],"proof":[20],"of":[21,57],"Bohr's":[22],"theorem,":[23],"which":[24],"asserts":[25],"that":[26,41],"its":[27],"majorant":[28],"$Mf(r)$":[30,54],"$1$":[34],"for":[35,53],"$0\\leq":[36],"r":[37],"\\leq":[38],"1/3$":[39],"and":[40],"$1/3$":[42],"optimal.":[44],"The":[45],"argument":[46],"also":[47],"yields":[48],"sharp":[50],"upper":[51],"bound":[52],"terms":[56],"$\\lvert":[58],"f(0)":[59],"\\rvert$,":[60],"complementing":[61],"result":[63],"due":[64],"to":[65],"Bombieri.":[66]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-25T00:00:00"}
