; TeX output 2001.05.23:1811 sbq;"V ff cmbx10R=UTGERSUNIVERSITY XGRADUATEPR=OGRAMINMATHEMATICSb<"V cmbx10WٚrittenQualifyingExamination K`y cmr10F*allUU2000,Day1 ThisexaminationwillbGegivenintwothree-hoursessions,`onetoGdayandoneto- morrow.*2At=yeachsessiontheexaminationwillhavetwoparts.*2AnswerallthreeofthequestionsinPartI(numbGered1{3)andthreeofthesixquestionsinPartIGI(numbered4{9).IfyouworkonmorethanthreequestionsinPartIGI,indicate"V cmbx10clearlywhichthreeshouldbGegraded.\Noadditionalcreditwillbegivenformorethanthreepartialsolutions.Ifqnothreequestionsareindicated,thentherstthreequestionsattemptedintheorderinwhichtheyappGearintheexaminationbook(s),andonlythose,willbGetheonesgraded. ;1 * sFirstTDa9y|PartI:Answereachofthefollowingthreequestions 1.j4Letb> cmmi10f 0er cmmi7n۲bGeasequenceofanalyticfunctionsontheopenunitdiscD,andcontinuousj4on theclosedunitdisc.]Assumefurhterthateac9hfndisinjectiveandthatfndconvergestoananalyticfunctionfuniformlyoncompactsubsetsofD.Provethateitherfisinjective,orthatUUfhisidenticallyaconstant.Note:qItUUisnotenoughtoquoteatheoremhere,afullargumentisneeded.2.UULetfڧ!", cmsy102L^O! cmsy71x[0;1].qDene,for0