[分析] 複變一題證明
Suppose that f is entire and that for each z, either |f(z)|≦1 or |f'(z)|≦1.
Prove that f is a linear polynomial.
Hint:Use a line integral to show
|f(z)| ≦ A + |z| where A = max{1,|f(0)|}
後面的簡答是寫:
z
Let f(z) = f(z_0) + ∫ f'(w) dw, where the path of integration is along the
z_0
ray from 0 to z, beginning at z_0, with z_0 = t_0 z, where
t_0 = sup{t_1: |f(tz)|≦1, 0≦t≦t_1}.
Clearly, then |f(z_0)|≦max{1,|f(0)|} and the integral is bounded by |z|.
=========
想問的是|f(z_0)|≦max{1,|f(0)|}
如果照t_0的假設 |f(0)|就會≦1, 是不是t_0的部分有錯
如果沒錯的話
可以分兩種情況1.|f(z_0)|≦1 2.|f'(z_0)|≦1
第二種情況要怎麼解決
z_0-δ z_0
我的想法是 |f(z_0)| = |f(0) + ∫ f'(w) dw + ∫ f'(w) dw|
0 z_0-δ
≦ |f(0)| + |B| + |C| where δ→0
|B| = |f(z_0-δ) - f(0)| = 0
不知道|C|要怎麼解決
z
然後 |∫ f'(w) dw| ≦|z| 這部分 是不是也要分兩種情況?
z_0
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 42.79.57.74
→
05/21 09:55, , 1F
05/21 09:55, 1F