In mathematics, Levinson's inequality is the following inequality, due to Norman Levinson, involving positive numbers. Let
a
>
0
and let
f
be a given function having a third derivative on the range
(
0
,
2
a
)
, and such that
f
‴
(
x
)
≥
0
for all
x
∈
(
0
,
2
a
)
. Suppose
0
<
x
i
≤
a
for
i
=
1
,
…
,
n
and
0
<
p
. Then
∑
i
=
1
n
p
i
f
(
x
i
)
∑
i
=
1
n
p
i
−
f
(
∑
i
=
1
n
p
i
x
i
∑
i
=
1
n
p
i
)
≤
∑
i
=
1
n
p
i
f
(
2
a
−
x
i
)
∑
i
=
1
n
p
i
−
f
(
∑
i
=
1
n
p
i
(
2
a
−
x
i
)
∑
i
=
1
n
p
i
)
.
The Ky Fan inequality is the special case of Levinson's inequality where
p
i
=
1
,
a
=
1
2
,
and
f
(
x
)
=
log
x
.
