背理法による。
|
z
−
z
0
|
<
δ
{\displaystyle |z-z_{0}|<\delta }
f
(
z
)
≠
{
a
,
b
}
{\displaystyle f(z)\neq \{a,b\}}
|
z
|
<
1
{\displaystyle |z|<1}
F
(
z
)
=
f
(
z
−
z
0
δ
)
−
a
b
−
a
≠
{
0
,
1
}
{\displaystyle F(z)={\frac {f\left({\tfrac {z-z_{0}}{\delta }}\right)-a}{b-a}}\neq \{0,1\}}
である。
M
=
sup
|
z
|
=
e
−
60
π
|
F
(
z
)
|
{\displaystyle M=\sup _{|z|=e^{-60\pi }}|F(z)|}
とする。
F
(
0
)
{\displaystyle F(0)}
カゾラーティ・ワイエルシュトラスの定理 により
∃
z
1
,
|
z
1
|
<
e
−
60
π
,
|
F
(
z
1
)
−
(
M
+
e
15
π
+
1
)
|
<
1
∃
z
2
,
|
z
2
|
<
|
z
1
|
,
|
F
(
z
2
)
−
1
|
<
1
2
{\displaystyle {\begin{aligned}&\exists {z_{1}},|z_{1}|<e^{-60\pi },\left|F(z_{1})-\left(M+e^{15\pi }+1\right)\right|<1\\&\exists {z_{2}},|z_{2}|<|z_{1}|,|F(z_{2})-1|<{\tfrac {1}{2}}\\\end{aligned}}}
が存在するので
F
1
(
t
)
=
f
(
z
2
e
59
π
i
t
)
F
2
(
t
)
=
log
F
1
(
t
)
2
π
i
,
(
|
ℑ
F
2
(
0
)
|
≤
1
)
F
3
(
t
)
=
F
2
(
t
)
G
(
t
)
=
sinh
−
1
F
3
(
t
)
,
(
|
ℑ
G
(
0
)
|
≤
π
)
{\displaystyle {\begin{aligned}&F_{1}(t)=f(z_{2}{e}^{59{\pi }it})\\&F_{2}(t)={\frac {\log {F_{1}(t)}}{2{\pi }i}},\qquad (|\Im {F_{2}(0)}|\leq 1)\\&F_{3}(t)={\sqrt {F_{2}(t)}}\\&G(t)=\sinh ^{-1}F_{3}(t),\qquad (|\Im {G(0)}|\leq \pi )\end{aligned}}}
とする。
|
z
2
|
<
|
z
1
|
<
e
−
60
π
{\displaystyle |z_{2}|<|z_{1}|<e^{-60\pi }}
F
1
(
t
)
{\displaystyle F_{1}(t)}
|
t
|
<
60
59
{\displaystyle |t|<{\tfrac {60}{59}}}
F
1
(
t
)
≠
{
0
,
1
}
{\displaystyle F_{1}(t)\neq \{0,1\}}
F
2
(
t
)
{\displaystyle F_{2}(t)}
F
2
(
t
)
≠
{
z
∈
Z
}
{\displaystyle F_{2}(t)\neq \{z\in \mathbb {Z} \}}
F
2
(
t
)
≠
{
0
,
1
}
{\displaystyle F_{2}(t)\neq \{0,1\}}
F
3
(
t
)
{\displaystyle F_{3}(t)}
F
3
(
t
)
≠
{
0
,
±
i
}
{\displaystyle F_{3}(t)\neq \{0,\pm {i}\}}
G
(
t
)
{\displaystyle G(t)}
G
(
t
)
≠
w
∈
{
sinh
−
1
n
+
i
π
m
|
(
n
,
m
)
∈
Z
2
}
{\displaystyle G(t){\neq }w\in \{\sinh ^{-1}{\sqrt {n}}+i\pi {m}|(n,m)\in \mathbb {Z} ^{2}\}}
である。従って、任意の
a
∈
C
{\displaystyle a\in \mathbb {C} }
G
(
t
)
+
a
−
w
{\displaystyle G(t)+a-w}
|
w
|
<
2
{\displaystyle |w|<2}
t
{\displaystyle t}
H
(
u
)
=
G
(
t
+
(
1
−
|
t
|
)
u
)
−
G
(
t
)
(
1
−
|
t
|
)
G
′
(
t
)
H
1
(
u
)
=
(
1
−
|
u
|
)
H
′
(
u
)
{\displaystyle {\begin{aligned}&H(u)={\frac {G\left(t+(1-|t|)u\right)-G(t)}{(1-|t|)G'(t)}}\\&H_{1}(u)=(1-|u|)H'(u)\\\end{aligned}}}
とする。
H
(
u
)
{\displaystyle H(u)}
|
u
|
<
60
59
{\displaystyle |u|<{\tfrac {60}{59}}}
H
(
u
)
+
a
−
w
{\displaystyle H(u)+a-w}
|
w
|
<
2
(
1
−
|
t
|
)
G
′
(
t
)
{\displaystyle |w|<{\frac {2}{(1-|t|)G'(t)}}}
H
1
(
0
)
=
1
{\displaystyle H_{1}(0)=1}
U
=
{
u
∈
C
:
|
u
|
<
1
,
(
1
−
|
u
|
)
|
H
′
(
u
)
|
≥
1
}
{\displaystyle \mathbb {U} =\{u\in \mathbb {C} :\;|u|<1,(1-|u|)\left|H'(u)\right|\geq {1}\}}
は空でない。
U
{\displaystyle \mathbb {U} }
u
1
{\displaystyle u_{1}}
J
(
v
)
=
2
(
H
(
1
−
|
u
1
|
2
v
+
u
1
)
−
H
(
u
1
)
)
{\displaystyle J(v)=2\left(H\left({\frac {1-|u_{1}|}{2}}v+u_{1}\right)-H\left(u_{1}\right)\right)}
とする。
J
(
v
)
{\displaystyle J(v)}
|
v
|
<
1
{\displaystyle |v|<1}
J
(
v
)
−
w
{\displaystyle J(v)-w}
|
w
|
<
4
(
1
−
|
t
|
)
G
′
(
t
)
{\displaystyle |w|<{\frac {4}{(1-|t|)G'(t)}}}
J
′
(
v
)
=
(
1
−
|
u
1
|
)
H
′
(
1
−
|
u
1
|
2
v
+
u
1
)
{\displaystyle J'(v)=(1-|u_{1}|)H'\left({\frac {1-|u_{1}|}{2}}v+u_{1}\right)}
となる。
|
J
′
(
0
)
|
=
|
H
1
(
u
1
)
|
=
1
{\displaystyle |J'(0)|=|H_{1}(u_{1})|=1}
|
J
′
(
v
)
|
{\displaystyle |J'(v)|}
最大値の原理 により
sup
|
v
|
<
1
|
J
′
(
v
)
|
≤
sup
|
u
|
=
1
+
|
u
1
|
2
(
1
−
|
u
|
)
|
H
′
(
u
)
|
≤
(
1
−
|
u
1
|
)
2
1
+
|
u
1
|
≤
2
{\displaystyle \sup _{|v|<1}|J'(v)|\leq \sup _{|u|={\tfrac {1+|u_{1}|}{2}}}(1-|u|)|H'(u)|\leq (1-|u_{1}|){\frac {2}{1+|u_{1}|}}\leq 2}
である。
|
J
′
(
v
)
−
1
|
≤
3
{\displaystyle \left|J'(v)-1\right|\leq 3}
シュワルツの補題 により
|
J
′
(
v
)
−
1
|
≤
3
v
{\displaystyle |J'(v)-1|\leq {3v}}
|
J
(
v
)
−
v
|
≤
3
2
|
v
|
2
{\displaystyle |J(v)-v|\leq {{\frac {3}{2}}|v|^{2}}}
となる。任意の
|
w
|
<
1
7
{\displaystyle |w|<{\tfrac {1}{7}}}
J
1
(
v
)
=
J
(
v
)
−
w
J
2
(
v
)
=
v
−
w
{\displaystyle {\begin{aligned}&J_{1}(v)=J(v)-w\\&J_{2}(v)=v-w\\\end{aligned}}}
とすれば
|
v
|
=
1
3
{\displaystyle |v|={\tfrac {1}{3}}}
|
J
1
(
v
)
−
J
2
(
v
)
|
=
|
J
(
v
)
−
v
|
≤
1
6
<
|
J
2
(
v
)
|
{\displaystyle |J_{1}(v)-J_{2}(v)|=|J(v)-v|\leq {\tfrac {1}{6}}<|J_{2}(v)|}
ルーシェの定理 により
J
1
(
v
)
{\displaystyle J_{1}(v)}
J
2
(
v
)
{\displaystyle J_{2}(v)}
|
v
|
<
1
3
{\displaystyle |v|<{\tfrac {1}{3}}}
J
2
(
v
)
{\displaystyle J_{2}(v)}
J
1
(
v
)
{\displaystyle J_{1}(v)}
|
4
(
1
−
|
t
|
)
G
′
(
t
)
|
≥
1
7
{\displaystyle \left|{\frac {4}{(1-|t|)G'(t)}}\right|\geq {\frac {1}{7}}}
でなければならない。
|
t
|
<
1
57
{\displaystyle |t|<{\tfrac {1}{57}}}
|
G
′
(
t
)
|
≤
57
2
{\displaystyle |G'(t)|\leq {\tfrac {57}{2}}}
|
F
1
(
0
)
|
=
|
F
(
z
2
)
|
<
1
2
{\displaystyle |F_{1}(0)|=|F(z_{2})|<{\tfrac {1}{2}}}
|
G
(
0
)
|
<
1
2
{\displaystyle |G(0)|<{\tfrac {1}{2}}}
|
G
(
t
)
|
≤
|
G
(
0
)
|
+
[
57
2
t
]
0
1
57
<
1
{\displaystyle |G(t)|\leq |G(0)|+\left[{\frac {57}{2}}t\right]_{0}^{\frac {1}{57}}<1}
|
F
1
(
t
)
|
=
|
e
2
π
sinh
2
G
(
t
)
|
<
e
2
π
e
2
<
e
15
π
{\displaystyle |F_{1}(t)|=\left|e^{2{\pi }\sinh ^{2}{G(t)}}\right|<{e^{2{\pi }e^{2}}}<e^{15\pi }}
となり
sup
|
z
|
=
|
z
2
|
|
F
(
z
)
|
≤
sup
|
x
|
≤
1
59
|
F
1
(
x
)
|
<
e
15
π
{\displaystyle \sup _{|z|=|z2|}|F(z)|\leq \sup _{|x|\leq {\tfrac {1}{59}}}|F_{1}(x)|<e^{15\pi }}
となるが
z
1
∈
{
z
∈
C
:
|
z
2
|
<
|
z
|
<
e
−
60
π
}
{\displaystyle z_{1}\in \{z\in \mathbb {C} :\;|z_{2}|<|z|<e^{-60\pi }\}}
であり
sup
|
z
|
=
e
−
60
π
|
F
(
z
)
|
=
M
<
|
F
(
z
1
)
|
{\displaystyle \sup _{|z|=e^{-60\pi }}|F(z)|=M<|F(z_{1})|}
であるから、最大値の原理 により
sup
|
z
|
=
|
z
2
|
|
F
(
z
)
|
≥
|
F
(
z
1
)
|
>
e
15
π
{\displaystyle \sup _{|z|=|z2|}|F(z)|\geq {|F(z_{1})|}>e^{15\pi }}
でなければならない。故に逆の仮定は矛盾を孕む。
![{\displaystyle |G(t)|\leq |G(0)|+\left[{\frac {57}{2}}t\right]_{0}^{\frac {1}{57}}<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e6b0f2e9b90aff3462f699ea46d04368e47f67)
