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Undergraduate contests
IMC
2021 IMC
7
7
Part of
2021 IMC
Problems
(1)
IMC 2021 P7: Maximum in the boundary multiplied by monic polynomial
Source: IMC 2021 P7
8/5/2021
Let
D
⊆
C
D \subseteq \mathbb{C}
D
⊆
C
be an open set containing the closed unit disk
{
z
:
∣
z
∣
≤
1
}
\{z : |z| \leq 1\}
{
z
:
∣
z
∣
≤
1
}
. Let
f
:
D
→
C
f : D \rightarrow \mathbb{C}
f
:
D
→
C
be a holomorphic function, and let
p
(
z
)
p(z)
p
(
z
)
be a monic polynomial. Prove that
∣
f
(
0
)
∣
≤
max
∣
z
∣
=
1
∣
f
(
z
)
p
(
z
)
∣
|f(0)| \leq \max_{|z|=1} |f(z)p(z)|
∣
f
(
0
)
∣
≤
∣
z
∣
=
1
max
∣
f
(
z
)
p
(
z
)
∣
complex analysis
IMC 2021
polynomial
Inequality