單葉函數的開始多項式

<正> §1 設k次對稱函數fk(z)=z+在單位圓|z|<1中正則單葉。記σ_n~((k))(z)=z+特別記σ_n~((1))(z)=σ_n(z). 舍苟證明一切σ_n(z)在圓|z|<1/4中單葉,且不能易以更大之數。列文

THE SECTIONS OF SCHLICHT FUNCTIONS

Let the k-symmetric function be regular and schlicht in the unit circle |z|<1. WriteSzego proves that all σ_n(z) are schlicht in the circle |z| <1/4.The present author establishes that all σ_n~((2))(z) are schlicht in the circle. The last result has been conjectured by, Joh. As for σ_n~((3)) (z), the problem has been studied by Ilief, and is solved completely in this note. We prove the following theorems:Theorem 1. All σ_n~((3))(z) are schticht in the circle The number can not be replaced by any large one.Comrbining the above results, we can stateTheorem 2. All σ_n~((k)) (z) are schlicht in the circle where k=1 2, 3. The number can not be replaced by any larger one.Let be regular and schlicht in the domain 1 < |ζ| < ∞. Denoting we can establish the followingTheorem 3. (i) If n>12,φ_n(ζ) is schlicht in the domain ∞>|ζ|≥(1-5lnn/n)~(-1/2).(ii) All φ_n(ζ) are schlicht in the domain ∞>|ζ|≥3/2.