圓環上的單葉函數

<正> 1.設G是z平面上的兩連區域,它的每一個境界部分都不止含有一點.我們知道有唯一的半徑R使圓環1<|ζ|

ON THE FUNCTIONS UNIVALENT IN A CIRCULAR RING

Let G be a doubly-connected domain in the z-plane. If, G can be represented conformally on a circular ring 1 <|ζ|0 and z=Φ_α(ζ) maps the ring 1<|ζ|R onto the unit circle|z|<1 with the slit 0≤z≤α.Theorem 4. Let D be a n-ply-connected domain in the ζ-plane, with the boundary continua γ_1, γ_2,…, γ_n.Let(ζ, z_o), z_o ∈D, be a function regular and univalent in D suchthat; (i) (z_o, z_o) =0, (ii) |(ζ, z_o)| = 1 for ζ∈γ_ν, (iii) arg (ζ, z_o) = const, forζ∈γ_μ,μ≠ν. Then, for any function regular and univalent in D,f(z_o)=0, |f(ζ)|≥1 onγ_ν, the inequalityholds true.