Carlemans olikhet är en matematisk olikhet namngiven efter Torsten Carleman , som var den förste att publicera olikheten 1923 [ 1]
Låt
a
1
,
a
2
,
.
.
.
{\displaystyle a_{1},a_{2},...}
följd av icke-negativa reella tal . Då gäller det att
∑
n
=
1
∞
(
a
1
.
.
.
a
n
)
1
n
≤
e
∑
n
=
1
∞
a
n
.
{\displaystyle \sum _{n=1}^{\infty }(a_{1}...a_{n})^{\frac {1}{n}}\leq e\sum _{n=1}^{\infty }a_{n}.}
Konstanten
e
{\displaystyle e}
a
1
,
a
2
.
.
.
{\displaystyle a_{1},a_{2}...}
Utgå från Hardys olikhet :
∑
n
=
1
∞
(
1
k
∑
k
=
1
n
a
k
)
p
<
(
p
p
−
1
)
p
∑
n
=
1
∞
a
n
p
{\displaystyle \sum _{n=1}^{\infty }\left({\frac {1}{k}}\sum _{k=1}^{n}a_{k}\right)^{p}<\left({\frac {p}{p-1}}\right)^{p}\sum _{n=1}^{\infty }a_{n}^{p}}
ta den inre summan i vänsterledet, ersätt
a
k
{\displaystyle a_{k}}
a
k
1
p
{\displaystyle a_{k}^{\frac {1}{p}}}
(
1
k
∑
k
=
1
n
a
k
1
p
)
p
=
exp
1
p
(
ln
∑
k
=
1
n
a
k
1
p
−
ln
∑
k
=
1
n
a
k
0
)
{\displaystyle \left({\frac {1}{k}}\sum _{k=1}^{n}a_{k}^{\frac {1}{p}}\right)^{p}=\exp {\frac {1}{p}}\left(\ln \sum _{k=1}^{n}a_{k}^{\frac {1}{p}}-\ln \sum _{k=1}^{n}a_{k}^{0}\right)}
Låt
p
→
∞
{\displaystyle p\to \infty }
x , som här är noll:
lim
p
→
∞
exp
1
p
(
ln
∑
k
=
1
n
a
k
1
p
−
ln
∑
k
=
1
n
a
k
0
)
=
exp
(
[
d
d
x
(
ln
∑
k
=
1
n
a
k
x
)
]
x
=
0
)
=
exp
(
[
∑
k
=
1
n
a
k
x
ln
a
k
∑
k
=
1
n
a
k
x
]
x
=
0
)
{\displaystyle \lim _{p\to \infty }\exp {\frac {1}{p}}\left(\ln \sum _{k=1}^{n}a_{k}^{\frac {1}{p}}-\ln \sum _{k=1}^{n}a_{k}^{0}\right)=\exp \left(\left[{\frac {d}{dx}}(\ln \sum _{k=1}^{n}a_{k}^{x})\right]_{x=0}\right)=\exp \left(\left[{\frac {\sum _{k=1}^{n}a_{k}^{x}\ln a_{k}}{\sum _{k=1}^{n}a_{k}^{x}}}\right]_{x=0}\right)}
Applicera nu
x
=
0
{\displaystyle x=0}
exp
(
[
∑
k
=
1
n
a
k
x
ln
a
k
∑
k
=
1
n
a
k
x
]
x
=
0
)
=
exp
(
1
n
∑
k
=
1
n
ln
a
k
)
=
(
∏
k
=
1
n
a
k
)
1
n
.
{\displaystyle \exp \left(\left[{\frac {\sum _{k=1}^{n}a_{k}^{x}\ln a_{k}}{\sum _{k=1}^{n}a_{k}^{x}}}\right]_{x=0}\right)=\exp \left({\frac {1}{n}}\sum _{k=1}^{n}\ln a_{k}\right)=\left(\prod _{k=1}^{n}a_{k}\right)^{\frac {1}{n}}.}
Betrakta nu högerledet i Hardys olikhet och utför samma steg, ersätt
a
k
{\displaystyle a_{k}\,}
a
k
1
p
{\displaystyle a_{k}^{\frac {1}{p}}}
p gå mot oändligheten
lim
p
→
∞
(
p
p
−
1
)
p
∑
n
=
1
∞
a
n
=
e
∑
n
=
1
∞
a
n
{\displaystyle \lim _{p\to \infty }\left({\frac {p}{p-1}}\right)^{p}\sum _{n=1}^{\infty }a_{n}=e\sum _{n=1}^{\infty }a_{n}}
detta ger oss den icke-strikta varianten av Carlemans olikhet:
∑
n
=
1
∞
(
∏
k
=
1
n
a
k
)
1
n
≤
e
∑
n
=
1
∞
a
n
{\displaystyle \sum _{n=1}^{\infty }\left(\prod _{k=1}^{n}a_{k}\right)^{\frac {1}{n}}\leq e\sum _{n=1}^{\infty }a_{n}}
^ T. Carleman, Sur les fonctions quasi-analytiques , Conférences faites au cinquième congres des mathématiciens Scandinaves, Helsinki (1923), 181-196.
![{\displaystyle \lim _{p\to \infty }\exp {\frac {1}{p}}\left(\ln \sum _{k=1}^{n}a_{k}^{\frac {1}{p}}-\ln \sum _{k=1}^{n}a_{k}^{0}\right)=\exp \left(\left[{\frac {d}{dx}}(\ln \sum _{k=1}^{n}a_{k}^{x})\right]_{x=0}\right)=\exp \left(\left[{\frac {\sum _{k=1}^{n}a_{k}^{x}\ln a_{k}}{\sum _{k=1}^{n}a_{k}^{x}}}\right]_{x=0}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e44099cd7dc44af3a273d3b3465a30f4c9fe70f5)
![{\displaystyle \exp \left(\left[{\frac {\sum _{k=1}^{n}a_{k}^{x}\ln a_{k}}{\sum _{k=1}^{n}a_{k}^{x}}}\right]_{x=0}\right)=\exp \left({\frac {1}{n}}\sum _{k=1}^{n}\ln a_{k}\right)=\left(\prod _{k=1}^{n}a_{k}\right)^{\frac {1}{n}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08d5a2ced747c02cce53eb381404d480dc6136e1)
