julias+lemma+and+bloch+constants

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1、万方数据SCIENCEINCHINA(SeriesA)Julia’slemmaandBlochconstantscH。NH。。i．。i(陈怀惠)，＆xl。NGc。。。舀i(￡袅继)算1DepartmentofMathematics．NanjingNormalUniversity．Nanjing210097．China；2DepartmentofMathematics．HuaiyinTeacher’sCollege．Huaiyin223001，ChinaCorrespondenceshouldbeaddressedtoChenHuaihui(email：hhchen@publ

2、iclpttjscn)ReceivedApril11．2002AbstractTheclassicalJulialslemmaisimproved．branchpointsofBlochfunctionsareinvestigated．andnewlowerboundsofBlochconstantsforfunctionswithbranchpointsareobtainedKeywords：analyticfunction，Julia’slemma，Blochfunction，BlochconstantLetD={。∈C：⋯<1)betheunitdiskontheco

3、mplexplaneC，letD(r)={2：㈦

4、diusofthelargestunivalentdiskcontainedinf(D)withcenter，(z)LetBi=snp{r(：，f)：2∈D}．ForapositiveintegerTt，let珥。(D)betheclassofallanalyticfunctions，onDsuchthatf’(2)=0impliesf”fz)一=，m’(2)=0TheBlochconstantfor月nisdefinedbyB。=inf{Bf：f∈上k(D)，，’(o)=1}Fortheabovenotions．seerefs『1—5]Afunctonf∈H1(D)isc

5、alledaBlochfunctionifsup{(1一z12)If’(。)：。∈D)<+ooBy8n，wedenotethesubclassofBlochfunctionswhichsatisfyconditionsf∈上k(D)，f(0)=0，，『(o)=1，and(1一H2)，”)I≤1for。∈DLandau[“showsthatB。=inf{Bf：f∈B。}AsmalllowerboundofBloch’sconstantB=B1canbeobtainedbyanelementarymethod[3，⋯In1938．Ahlfors[6，7】generalizedt

6、heSchwarz—Picklemmawithwhichheobtainedaremarkablelowerestimate：B≥、／3／4≈o4330．Atthesametime，AhlforsandGrunsky{“gaveanimportantexampletoestablishanupperboundofB，whichisexpressedbyGammafunctionsandequalto0．4719approximatelyPeopleconjecturedthatthisupperboundistheprecisevalueofBloch’sconstantI

7、tisafamousandextremelyhardproblemtofindtheexactvalueofBloch’sconstantForB。，n>1，Ahlfors’methodalsogavealowerbound：Bn≥、／n(n+2)／(2(n+1))TheirupperboundswereinvestigatedbyMinda{51Overalongperiodoftime，noessentialachievementhadbeenattainedinthistopicuntilBonk’swork