关于单叶函数系数之—基本引理及其应用

<正> 设函数 f(z)=z+a_2Z~2+…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)|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

SOME INEQUALITIES DERIVED FROM FUNDAMENTAL LEMMA CONCERNING SCHLICHT FUNCTIONS

Let S be the class of functions f(Z)=Z+a_2Z~2+… regular and schlichtin the unit circle |Z|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 a_1=0,the equality(?)holds when and only when(?)Theorem 2.If F(ξ)∈Σ,then(?)where(?)and x_0=0,92402…is the positive root of the equation 5x~3+27x~2-27=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(x_1,…,X_N;y_1,…,y_N)withx_n+iy_n=a_n,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.Golusin3]whenM→∞.