MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
1959 Miklós Schweitzer
9
9
Part of
1959 Miklós Schweitzer
Problems
(1)
Miklós Schweitzer 1959- Problem 9
Source:
11/8/2015
9. Let
f
(
z
)
=
z
n
+
a
1
z
n
−
1
+
⋯
+
a
n
f(z)= z^n +a_1 z^{n-1}+\dots + a_n
f
(
z
)
=
z
n
+
a
1
z
n
−
1
+
⋯
+
a
n
be a polynomial over the field of the complex numbers and let
E
f
E_f
E
f
denote the closed (not necessarily connected) domain of complex numbers
z
z
z
for which
∣
f
(
z
)
∣
≤
1
\mid f(z) \mid \leq 1
∣
f
(
z
)
∣≤
1
. Show that there exists a point
z
0
∈
E
f
z_0 \in E_f
z
0
∈
E
f
such that
∣
f
′
(
z
0
)
∣
≥
n
\mid f'(z_0) \mid \geq n
∣
f
′
(
z
0
)
∣≥
n
. (F. 5)
college contests