{"id":"https://openalex.org/W7221752425","doi":"https://doi.org/10.48550/arxiv.2610.15067","title":"Maximum-modulus points on concentric circles","display_name":"Maximum-modulus points on concentric circles","publication_year":2026,"publication_date":"2026-10-10","ids":{"openalex":"https://openalex.org/W7221752425","doi":"https://doi.org/10.48550/arxiv.2610.15067"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.15067","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.15067","pdf_url":"https://arxiv.org/pdf/2610.15067","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/2610.15067","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5158756794","display_name":"Yunpeng Li","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210116924","display_name":"Chinese University of Hong Kong, Shenzhen","ror":"https://ror.org/02d5ks197","country_code":"CN","type":"education","lineage":["https://openalex.org/I177725633","https://openalex.org/I180726961","https://openalex.org/I4210116924"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Li, Yunpeng","raw_affiliation_strings":["School of Data Science , The Chinese University of Hong Kong , Shenzhen , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Data Science , The Chinese University of Hong Kong , Shenzhen , China","institution_ids":["https://openalex.org/I4210116924"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5158756794"],"corresponding_institution_ids":["https://openalex.org/I4210116924"],"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/T12812","display_name":"Meromorphic and Entire Functions","score":0.9955999851226807,"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/T12812","display_name":"Meromorphic and Entire Functions","score":0.9955999851226807,"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/T11822","display_name":"Analytic and geometric function theory","score":0.0010000000474974513,"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":0.000699999975040555,"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/entire-functions","display_name":"entire functions","score":0.9279999732971191},{"id":"https://openalex.org/keywords/concentric-circles","display_name":"concentric circles","score":0.906000018119812},{"id":"https://openalex.org/keywords/logarithmic-derivative","display_name":"logarithmic derivative","score":0.6639999747276306},{"id":"https://openalex.org/keywords/critical-points","display_name":"critical points","score":0.8399999737739563},{"id":"https://openalex.org/keywords/local-parametrization","display_name":"local parametrization","score":0.7289999723434448},{"id":"https://openalex.org/keywords/analytic-continuation","display_name":"analytic continuation","score":0.7839999794960022},{"id":"https://openalex.org/keywords/sharp-constants","display_name":"sharp constants","score":0.4519999921321869}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8327000141143799},{"id":"https://openalex.org/C90296322","wikidata":"https://www.wikidata.org/wiki/Q619776","display_name":"Concentric","level":2,"score":0.7207000255584717},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.58160001039505},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.5192999839782715},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4948999881744385},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4659999907016754},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.45210000872612},{"id":"https://openalex.org/C65682993","wikidata":"https://www.wikidata.org/wiki/Q1056451","display_name":"Reflection (computer programming)","level":2,"score":0.42989999055862427},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4187999963760376},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.4133000075817108},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.41190001368522644},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.39309999346733093},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.3521000146865845},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.33820000290870667},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.3000999987125397},{"id":"https://openalex.org/C140528856","wikidata":"https://www.wikidata.org/wiki/Q191884","display_name":"Inverse function","level":3,"score":0.29750001430511475},{"id":"https://openalex.org/C6128204","wikidata":"https://www.wikidata.org/wiki/Q847600","display_name":"Entire function","level":2,"score":0.296099990606308},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28040000796318054},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.27970001101493835},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.25369998812675476}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.15067","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.15067","pdf_url":"https://arxiv.org/pdf/2610.15067","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.2610.15067","doi":"https://doi.org/10.48550/arxiv.2610.15067","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.15067","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:2610.15067","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.15067","pdf_url":"https://arxiv.org/pdf/2610.15067","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/W7221752425.pdf","grobid_xml":"https://content.openalex.org/works/W7221752425.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W29624908","https://openalex.org/W1544790203","https://openalex.org/W4386700998","https://openalex.org/W7168184216"],"related_works":["https://openalex.org/W4386700998","https://openalex.org/W7222786763","https://openalex.org/W4410061886","https://openalex.org/W1968006191","https://openalex.org/W7168184216","https://openalex.org/W2069086707","https://openalex.org/W4411522842","https://openalex.org/W2015717104","https://openalex.org/W1970155810","https://openalex.org/W1987395052"],"abstract_inverted_index":{"We":[0,56],"give":[1],"a":[2,21,47,76,86,93],"negative":[3],"answer":[4],"to":[5,29],"the":[6,14,42,51,67],"second":[7],"question":[8],"of":[9,17,20,54,85,96],"Erd\u0151s":[10],"Problem":[11],"#1117:":[12],"can":[13],"number":[15],"$\u03bd_f(r)$":[16],"maximum-modulus":[18],"points":[19],"nonzero,":[22],"non-monomial":[23],"entire":[24],"function":[25],"on":[26],"$|z|=r$":[27],"tend":[28],"infinity":[30],"as":[31],"$r\\to\\infty$?":[32],"Write":[33],"$f(z)=z^m":[34],"g(z)$":[35],"with":[36,46,89],"$g(0)\\ne0$,":[37],"and":[38,65,92],"let":[39],"$k$":[40],"be":[41],"first":[43],"positive":[44],"exponent":[45],"nonzero":[48],"coefficient":[49],"in":[50,63],"Taylor":[52],"expansion":[53],"$g$.":[55],"prove":[57],"that":[58,66],"$\\{r>0:\u03bd_f(r)>2k\\}$":[59],"is":[60,70],"locally":[61],"finite":[62,82],"$(0,\\infty)$,":[64],"constant":[68],"$2k$":[69],"sharp.":[71],"In":[72],"particular,":[73],"$\\liminf_{r\\to\\infty}\u03bd_f(r)\\le2k<\\infty$,":[74],"without":[75],"growth":[77],"assumption.":[78],"The":[79],"proof":[80],"combines":[81],"local":[83,94],"parametrizations":[84],"logarithmic-derivative":[87],"correspondence":[88],"analytic":[90],"continuation":[91],"description":[95],"stationary":[97],"points.":[98]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-15T00:00:00"}
