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Miklós Schweitzer
1958 Miklós Schweitzer
9
Miklós Schweitzer 1958- Problem 9
Miklós Schweitzer 1958- Problem 9
Source:
October 23, 2015
college contests
Problem Statement
9. Show that if
f
(
z
)
=
1
+
a
1
z
+
a
2
z
2
+
…
f(z) = 1+a_1 z+a_2z^2+\dots
f
(
z
)
=
1
+
a
1
z
+
a
2
z
2
+
…
for
∣
z
∣
≤
1
\mid z \mid\leq 1
∣
z
∣≤
1
and
1
2
π
∫
0
2
π
∣
f
(
e
i
ϕ
)
∣
2
d
ϕ
<
(
1
+
∣
a
1
∣
2
4
)
2
\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2
2
π
1
∫
0
2
π
∣
f
(
e
i
ϕ
)
∣
2
d
ϕ
<
(
1
+
4
∣
a
1
∣
2
)
2
,then
f
(
z
)
f(z)
f
(
z
)
has a root in the disc
∣
z
∣
≤
1
\mid z \mid \leq 1
∣
z
∣≤
1
.(F. 4)
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