; TeX output 2002.09.05:1551 } hGS 9K`y UU cmr10AnnalesAcademiScienĽtiarumFN8ennic MathematicaVN8olumen27,2002,381{390"荑@7"V cmbx10ONZEROSOFNORMALFUNCTIONS X"V cmbx10MariaTNo9wakQcUniwĽersytetMariiCurie-Sk glo;Bdowskiej,InstytutMatematyki Qpl.MariiCurie-Sk glo;BdoĽwskiej1,PL-20-031Lublin,Poland;nowakm@golem.umcs.lublin.plD"V UU cmbx10DAbstract."WN8egivĽenecessaryconditionsforzerosetsofnormalfunctions,5Klittlenormal functionsandfunctionsofuniformlyb;BoundedcĽharacteristic.W 1.pIn9troQduction K`y cmr10Ar-functionr5b> cmmi10fīmeromorphicr5intheunitdiscD8=!", cmsy10fzgϷ2C:jzpj<1gisr5normalif䍍(1:1)zmkfk=r5supi 0er cmmi7zI{O! cmsy72f$ cmbx7D+(18 jzpjٓR cmr72|s)f](z)<1;wwhereUUf^](zp)=jf^0ȫ(z)j=bu cmex10 W18+jf(z)j^2|sb ʫ.qIfZ(1:2) Ԟlimꪍ ]jzI{j!1 .۫(18 jzpj2|s)f](z)=0;"Uthenf5iscalledalittlenormalfunction.vTheclassofnormalandlittlenormalfuctionsUUwillbGedenotedbyUUNlpandN0|s,UUrespectively*.TheUUBloGchspaceUU2}>y rsfs10BconsistsofthosefunctionsfhanalyticonDforwhich supi شzI{2D jf0ȫ(zp)j(18 jzj2|s)<1;wsandUUthelittleBloGchspaceUUB0ȫconsistsofthosefunctionsfڧ2Bforwhich Jlimꪍ 8jzI{j!1 /jf0ȫ(zp)j(18 jzj2|s)=0: Sincejf^](zp)jjf^0ȫ(z)j ,itisclearthateveryBloGchfunctionisanormalfunction..ItwasalsoobservedbyTse[T]thatifѱfڧ2Bh,theng"=e^f82N.=ThereareextensiveresultsabGoutnormalandBlochfunctions,see,e.g.,[L],[ACP]andreferencesgiventhere.In"1988D.Ulrich[U]usedrandomseriestoshowthatzerosetsofelementsof\B09are\dierentfromthezerosetsofelementsof\Bh.Moreexactly*,^heprovedthatCthereisafunctioninCBcwhosezeroscannotbGezerosofanyfunctioninBor}. ! ff xsUU2000MathematicsSubvjectClassication:%Primary30D35,30D45. ~ * 382 !p0J cmsl10MariaUUNowakhGIfJfznq~gisJthesequenceofzerosofanormalfunctionJf]andjz1|sj_ jz2j L<1,,UUthenwecallUUfznq~gtheUUorderedzerosofUUf.\RecentlyithasbGeenprovedin[GNW]thatiffznq~garetheorderedzerosofaBloGchUUfunctionnonvqanishingatzero,then"N(1:3)eNcYaǴn=1<$y;B1s#w fe ? (֍jznq~j =OGb t(logTN)1=2ʟb'!;as!DZN3!1;"AэandUUifUUfznq~g( zn86=0)aretheorderedzerosofalittleBloGchfunction,then"g~&NdpYc>n=1<$z¹1uow fe ? (֍jznq~j $=ob W(logTN)1=2ʟb'!;as!DZN3!1:ThisresulthasbGeenmotivqatedbyCh.Horowitz'papGer[Hor]onzerosoffunctions inrBergmanspacerA^pR.& Horowitzprovedthatifrfznq~garerorderedzerosofafunctioninUUA^pR,0