Bloch函数和小Bloch函数判别准则的推广第24卷第3期2006年8月贵州师范大学(自然科学版)JournalofGuizhouNormalUniversity(NaturalSciences)Vo1.24.No.3Aug.2006:1004--5570(2006)03—0o84—04Ageneralizati0nofafunctionsandlittlecriterionofBlochBlochfunctionsYANGCong—li.WUPeng—cheng(SchoolofMathematicsandComputerScience,GuizhouNormalUniversity,Guiyang,Guizhou550001,China)Abstract:Inthispaper,weinvestigatethecriterionofBlochfunctionandlittleBlochfunction,weob—tainanextendedresultsofAulaskari—Lappan,Minda,Aulaskari—WulanandWu.Keywords:Blochfunction;littleBloehfunctionBloch函数和小Bloch函数判别准则的推广杨丛丽,伍鹏程(贵州师范大学数学与计算机科学学院,贵州贵阳550001)摘要:研究了Bloch函数和小Bloch函数的判别准则,并且推广了Aulaskari-lappan,Minda,Aulaskari-Wulan和Wu的结果.关键词:Bloch函数;小Bloch函数:0174文献标识码:A1IntroductionLetA={z:lzl《1}betheunitdiskinthecomplexplane.ananalyticfunctionfintheunitdiskAiSsaidtobeaBlochfunction.ifitsatisfiessuP(I—Izl)l厂z)l《∞.AnanalyticfunctionfintheunitdiskAiscalledalittleBloehfunctioniflim.+l(1一Izl)If(z)l=0-AulaskariandLappan[】andMinda[】gaveanMtemafivecharacterizationofBloehfunctions.TheyprovedthefollowingTheoremALM:AfunctionfanalyticintheU—nitdiskAisnotaBlochfunctionifandonlyifthereexistasequence{z}cAwithlzl---~1,andase—quence{P}ofpositivenumberssatisfying_,suchthatthesequencetf(z+p)-f()}convergeslocallyuniformlytoanonconstantanalyticfunctioninC.ForlittleBloehfunctions,AulaskariandWu-lan[】provedthe~llowing收稿日期:2006—03—28基金项目:ResearchsupportedbythegovernorfoundationofGuizhouprovince[02No.1]andGuizhouspecialoutstandingscientistfoundation.作者简介:YangCongli(1981一),Female,bominLiping,Guizhou,graduatestudent,majorintheo~offunction.84第3期杨丛丽,伍鹏程:Bloeh函数和小Bloeh函数判别准则的推广TheoremAW:AfunctionfanMyticontheunitdiskAisnotahttleBlochfunctionifandonlyifthereexistaconstantR》0,asequence{:}cAwithIzI---~1,andasequence{P}ofpositivenum—bersSatisng《1,suchtllatthese—quence{,(:+p)-f(z)}convergeslocallyuni—formlytoanonconstantanalyticfunctioninIl《R.2MainresultandProofWeinvestigatethepossibilityofintroducingaslidingscaleinvolvingaparameterintothesetheoremsinanalogytoPang'Sgeneralization['】ofZalcman'SLemmat51.Letbeagivenrealnumberwitll0≤《land0《《1f0rBloehfunctionsandlittleBloehfunctions.Wugetasimilarcriterion[引.Inthispa—per,weuseWu'Sideaandobtainanextendedresultinthefollowing.Theorem1.LetfbeananalyticfunctionintheunitdiskA.and仅a~ivenrealnumberwith一1《仅《1.ThenfisnotaBlochfunctionifandonlyifthereexistasequence{:}cAwithIzI_+1,andasequence{P}ofpositivenumberssatisfying一+o,suchn1esequence}convergeslocallyuniformlytoinC,whereaisaconstantwithlal=1.Proof.,I1leprooffollowsthepaRemin[6].As—samethatfisnotaBloehfunction.Thenthereexistsasequence{:}cAwithl_+1suchthat(1一l)l厂(::)l_+∞,as,l_+∞.(1)1wecanassumel:l》÷forall,l∈N.Let,万∈-《and-=rn+-==,(1一2)I():(1一)(1厂()1)r"SinceIz"I《rnandIz-I1一——一=rn(3+Iz-I)(1一Iz-I)4(1+Iz-I)ThUS(rn+Iz"I)(r^一Iz"I)rn≥:(1一)(1厂()1)≥(1一)(tf(:)1)≥(÷)(1一?(tf(:)I)_+∞.NOWset(一广=(南).Inparticular,,IzI,rn—n'南Therefore,thefunctionsg():aI.ede6nedonthediskll《:D.(2)(3)(4)(5)(6)(7)(8):=R_+∞.Hence,accordingto(2)and(5),foreachfixedR》O,whenIl≤≤,wehavelg()I=p1一'tf(z+p)I(1厂(+p)1)(1一)(1一)rnrn(If(1)(1一Iz.一I2)(1一)rnr"=黼rII≤n一n一一+pl85贵州师范大学(自然科学版)第24卷r^+Iz1一Ir^+Iz1+pRr一Iz1...