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关于单叶函数系数之—基本引理及其应用

SOME INEQUALITIES DERIVED FROM FUNDAMENTAL LEMMA CONCERNING SCHLICHT FUNCTIONS

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【作者】 刘醴泉

【Author】 LIU LI-CHUAN(Futan University)

【机构】 复旦大学

【摘要】 <正> 设函数 f(z)=z+a2Z2+…在单位圆|z|<1上是正则的单叶的.这种函数的全体形成一族 S.S 中满足条件|f(z)|<M的函数的全体形成 S 的一个子族 SM.设函数(?)在单位圆的外部区域|ζ|>1上是单叶的,除开极点ζ=∞是正则的.这种函数的全体形成一族∑.∑中满足条件|F(ζ)|>R的函

【Abstract】 Let S be the class of functions f(Z)=Z+a2Z2+… regular and schlichtin the unit circle |Z|<l;and SM be the subclass of functions f(Z)of Ssuch that |f(Z)|<M for|Z|<l.Let Σ be the class of functions(?)regular and schlicht in the domain l<|ξ|<∞;and ΣRbe the subclass of functions F(ξ)of Σ such that|F(ξ)|<M for|ξ|>l.The chief object of this work is to establish the following two theorems.Theorem 1.If f(ξ)∈Σ,thenThe sign of equality holds only whenIn the case a1=0,the equality(?)holds when and only when(?)Theorem 2.If F(ξ)∈Σ,then(?)where(?)and x0=0,92402…is the positive root of the equation 5x3+27x2-27=0,Equality holds only for(?)The proofs of theorem 1 and theorem 2 are based on a fundamentallemma concerning the extremum property of the functional H(x1,…,XN;y1,…,yN)withxn+iyn=an,n=1,…,Nwe prove also the following two theorems.Theorem 3.If(?)then(?)the estimates are sharp.Theorem 4.If(?)then(?)2n+lthe estimates are sharp.These propositions are reduced to the theorems of G.M.Golusin[3]whenM→∞.

【关键词】 基本引理单位圆外部区域子族一侗酿泉const完全平方沙玛三根
  • 【文献出处】 数学学报 ,Acta Mathematica Sinica , 编辑部邮箱 ,1957年02期
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