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1、SharpformsofNevanlinnaerrortermsindifferentialequationsRistoKorhonenAbstractSharpversionsofsomeclassicalresultsindifferentialequationsaregiven.MainresultsconsistsofaClunieandaMohon’kotypetheorems,bothwithsharpformsoferrorterms.Thesharpnessoftheseresultsisdiscussedandsomeapplicationst
2、ononlineardifferentialequationsaregivenintheconludingremarks.Moreover,ashortintroductionontheconnectionbetweenNevanlinnatheoryandnumbertheory,aswellasontheirrelationtodifferentialequations,isgiven.Inaddi-tion,abriefreviewontherecentdevelopmentsinthefieldofsharperrortermanalysisispresen
3、ted.1IntroductionRolfNevanlinna’stheoryofvaluedistributionisundoubtedlyoneofthegreatmathematicaldiscoveriesofthetwentiethcentury.Nevanlinnalaidthefoundationsofthetheoryina100pagesarticleappearedin1925[15].ThisremarkablecontributionwaslaterondescribedbyHermannWeylas...oneofthefewgrea
4、tmathematicaleventsinourcentury[24].Theonlyremainingbigopenquestion,proposedandpartiallysolvedbyNevanlinnahimself,wasforalongtimetheNevanlinnainverseproblem,whichisessentiallyaproblemoffindingameromorphicfunctionwithprescribeddeficientvalues.Thisproblemwassolvedin1976byDavidDrasin,who
5、usedquasiconformalmappingstoconstructthedesiredfunction[5].arXiv:math/0608514v1[math.CV]21Aug2006Undoubtedlythisresultwasasubstantialcontributiontothetheory,andsomewouldevengosofarastosaythatDrasin’sworkfinallycompletedNevanlinnatheory.But,nodoubt,therichnessofvaluedistributiontheory
6、goesmuchfurtherthanthat.Notonlyithasnumerousapplicationsinthefieldsofdifferentialandfunctionalequations,butthereisaprofoundrelationbetweenNevanlinnatheoryandnumbertheory,whichalsoextendstothetheoryofdifferentialequations.Inthispaperwewillconcentrateonthisdeepconnectionbetweenthesethree
7、theories.Thepresentpaperisorganizedasfollows.WestartbyrecallingthenecessarynotationinSection2.WethencontinuebygivingashortintroductiontothedeepconnectionbetweenNevanlinnatheoryandnumbertheoryinSection3.Thisisbynomeanscompletereviewofthetopic,and,thereforewereferto[1],[3],[18]and[23]
8、foramorecomprehensi
