; TeX output 2002.09.05:1552 hGS 9K`y UU cmr10AnnalesAcademiScienĽtiarumFN8ennic MathematicaVN8olumen27,2002,257{272 Rs7"V cmbx10ADISTORٚTIONTHEOREMFORAmXBOUNDEDUNIVwxALENTFUNCTIONS "V cmbx10Oliv9erTRothhMUnivĽersitUatW;Curzburg,MathematischesInstitut EQD-97074W;Curzburg,GermanĽy;roth@mathematik.uni-wuerzburg.deD"V UU cmbx10DAbstract.s}WN8eproĽveadistortiontheoremforb;Boundeduniv{alentfunctions.s}Ourresult includesyandrenesdistortiontheoremsduetoKo;Bebe,PicĽk,Blatter,KimyandMinda,MaandMinda,andJenkins.n 1.pIn9troQduction7 K`y cmr10Recently*,usingthegeneralcoGecienttheorem,Jenkins[6]provedthesharpestimate-8(1b> cmmi10:1)5/!", cmsy10jf(zٓR cmr71|s)8 f(z2|s)j<$sinh*2%Kw fe Cr V2(2coshG2p%)r1 0er cmmi7=pI#bu cmex10 MGjD1f(z1)jp2+8jD1f(z2)jpRb 4߮1=pforYanyfunctionYfm2analyticandunivqalentintheunitdiskYDE:=fz>ܷ2Cjjzpj<1gand۩any۩p1 ,=where%denotes۩thehypGerbolic۩distancedf$ cmbx7DeT(z1|s;z2)bGetweenz1andz2obtainedfromthelineelementjdzpj=(1 jzj^2|s) ,2andD1f(z)=(1 jzj^2|s)f^O! cmsy70ȫ(z)isithe\hypGerbolic"iderivqativeofif.Jenkinsalsoshowedthatinequality(1.1)isnotUUtrueforUU0