Quest articol chi l'è scrivuu in Koiné occidentala .
Reguredemm che par cada funziun intrega
g
{\displaystyle g}
M
r
(
g
)
:=
max
|
z
|
=
r
{
|
g
(
z
)
|
}
{\displaystyle M_{r}(g):=\max _{\vert z\vert =r}\{\vert g(z)\vert \}}
LEMA : Al síes
K
{\displaystyle {\mathcal {K}}}
cungjuunt di funziun ulumòorf
h
:
D
(
0
,
1
)
→
C
{\displaystyle h:\mathbb {D} (0,1)\to \mathbb {C} }
h
(
0
)
=
0
{\displaystyle h(0)=0}
M
1
/
2
(
h
)
≥
1
{\displaystyle M_{1/2}(h)\geq 1}
h
{\displaystyle h}
c
(
h
)
:=
sup
{
r
>
0
:
∂
D
(
0
,
r
)
}
⊂
h
(
D
(
0
,
1
)
)
{\displaystyle c(h):=\sup\{r>0:\partial \mathbb {D} (0,r)\}\subset h(\mathbb {D} (0,1))}
inf
{
c
(
h
)
:
h
∈
K
}
>
0
{\displaystyle \inf\{c(h):h\in {\mathcal {K}}\}>0}
Demustrazziun : süpusemm par l'assüürt ch'al esiist una sequenza
{
h
n
}
⊂
K
{\displaystyle \{h_{n}\}\subset {\mathcal {K}}}
lim
n
→
∞
c
(
h
n
)
=
0
{\displaystyle \lim _{n\to \infty }c(h_{n})=0}
n
{\displaystyle n}
D
(
0
,
1
)
{\displaystyle \mathbb {D} (0,1)}
D
(
0
,
1
/
2
)
{\displaystyle \mathbb {D} (0,1/2)}
D
(
0
,
1
/
4
)
{\displaystyle \mathbb {D} (0,1/4)}
h
n
(
D
(
0
,
1
)
)
{\displaystyle h_{n}(\mathbb {D} (0,1))}
{
h
n
}
{\displaystyle \{h_{n}\}}
n
{\displaystyle n}
h
n
(
D
(
0
,
1
)
)
{\displaystyle h_{n}(\mathbb {D} (0,1))}
a
n
,
b
n
,
c
n
{\displaystyle {a_{n},b_{n},c_{n}}}
min
{
|
a
n
−
b
n
|
,
|
b
n
−
c
n
|
,
|
c
n
−
a
n
|
}
≥
1
/
4
{\displaystyle \min\{\vert a_{n}-b_{n}\vert ,\vert b_{n}-c_{n}\vert ,\vert c_{n}-a_{n}\vert \}\geq 1/4}
n
{\displaystyle n}
teurema da Montel
{
h
n
}
{\displaystyle \{h_{n}\}}
{
h
n
}
{\displaystyle \{h_{n}\}}
h
:
D
→
C
{\displaystyle h:\mathbb {D} \to \mathbb {C} }
lema da Hurwitz
h
∈
K
{\displaystyle h\in {\mathcal {K}}}
h
{\displaystyle h}
r
>
0
{\displaystyle r>0}
∂
D
(
0
,
r
)
⊂
h
(
D
(
0
,
1
/
2
)
)
{\displaystyle \partial \mathbb {D} (0,r)\subset h(\mathbb {D} (0,1/2))}
lema da Hurwitz
a s'uteegn, par cada
n
{\displaystyle n}
c
(
h
n
)
≥
r
{\displaystyle c(h_{n})\geq r}
