【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 S_M.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的函更多还原

【Abstract】 Let S be the class of functions f（Z）=Z+a₂Z²+… regular and schlichtin the unit circle |Z|<l;and S_M 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 Σ_Rbe 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,…,X_N;y₁,…,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.Golusin[3]whenM→∞.更多还原