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In mathematics , Mahler's inequality , named after Kurt Mahler , states that the geometric mean of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means:
∏
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=
1
n
(
x
k
+
y
k
)
1
/
n
≥
∏
k
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n
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n
+
∏
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n
y
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/
n
{\displaystyle \prod _{k=1}^{n}(x_{k}+y_{k})^{1/n}\geq \prod _{k=1}^{n}x_{k}^{1/n}+\prod _{k=1}^{n}y_{k}^{1/n}}
when x k y k k .
Proof By the inequality of arithmetic and geometric means , we have:
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k
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1
n
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x
k
x
k
+
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k
)
1
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n
≤
1
n
∑
k
=
1
n
x
k
x
k
+
y
k
,
{\displaystyle \prod _{k=1}^{n}\left({x_{k} \over x_{k}+y_{k}}\right)^{1/n}\leq {1 \over n}\sum _{k=1}^{n}{x_{k} \over x_{k}+y_{k}},}
and
∏
k
=
1
n
(
y
k
x
k
+
y
k
)
1
/
n
≤
1
n
∑
k
=
1
n
y
k
x
k
+
y
k
.
{\displaystyle \prod _{k=1}^{n}\left({y_{k} \over x_{k}+y_{k}}\right)^{1/n}\leq {1 \over n}\sum _{k=1}^{n}{y_{k} \over x_{k}+y_{k}}.}
Hence,
∏
k
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1
n
(
x
k
x
k
+
y
k
)
1
/
n
+
∏
k
=
1
n
(
y
k
x
k
+
y
k
)
1
/
n
≤
1
n
n
=
1.
{\displaystyle \prod _{k=1}^{n}\left({x_{k} \over x_{k}+y_{k}}\right)^{1/n}+\prod _{k=1}^{n}\left({y_{k} \over x_{k}+y_{k}}\right)^{1/n}\leq {1 \over n}n=1.}
Clearing denominators then gives the desired result.
See also References
