[複變] 兩題複變請教

[wayne0824 (萊恩)]

以下兩題複變請教><

1.A Jordan curve in C can be considered as a subset of C homeomorphic to the
  unit circle. Now let Γ⊂C be a jordan cuvre and let φ:Γ→∂D be a
  homeomorphism. Show that, given ε>0, there exists δ>0 such that
  if a,b∈Γ and |a-b|<δ, then

  (a)One of the two circular arcs determined by φ(a) and φ(b) has strictly
     shorter length;

  (b)The diameter of σ_ab, the image of this shorter arc under φ^-1,
     is less than ε.



2.Let Ω be a bounded domain an let f:H+→Ω be bi-holomorphic.
  Let r>0 and ω_r(t)=f(re^it), 0<t<π and l(ω_r)=length of ω_r.
  Show that

  (a)ω_r(0+)=a and ω_r(π-)=b for some a,b∈Γ if l(ω_r)<∞.

  (b)There exist r_n↘0 such that l(ω_r_n)→0 as n→0.
     Hint: Consider the fact that Area(f(D_r(0)∩H+))≦Area(Ω) for all
           r>0, and use the integral form of Cauchy-Schwarz inequality.

  (c)Choose a sequence r_n↘0 such that l(ω_r_n)→0 and set
     ω_r_n(0+)=a_n, ω_r_n(π-)=b_n. Let Ω_n be the inside of the Jordan
     curve ω_r_n∪σ_anbn. Then diam(Ω)→0.


以上兩題(雖然有很多小題)@@
請教各位了，希望附上過程，拜託><

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推 HeCker      : 你可以參考Caratheodory theorem的證明
→ HeCker      : 第二題的H+是上半平面?
→ HeCker      : 沒事 請忽略我的第二句話..