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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Let
a
1
,
a
2
,
a
3
,
…
{\displaystyle a_{1},a_{2},a_{3},\dots }
be a sequence of non-negative real numbers , then
∑
n
=
1
∞
(
a
1
a
2
⋯
a
n
)
1
/
n
≤
e
∑
n
=
1
∞
a
n
.
{\displaystyle \sum _{n=1}^{\infty }\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}\leq \mathrm {e} \sum _{n=1}^{\infty }a_{n}.}
The constant
e
{\displaystyle \mathrm {e} }
(euler number) in the inequality is optimal, that is, the inequality does not always hold if
e
{\displaystyle \mathrm {e} }
is replaced by a smaller number. The inequality is strict (it holds with "<" instead of "≤") if some element in the sequence is non-zero.
Integral version [ edit ]
Carleman's inequality has an integral version, which states that
∫
0
∞
exp
{
1
x
∫
0
x
ln
f
(
t
)
d
t
}
d
x
≤
e
∫
0
∞
f
(
x
)
d
x
{\displaystyle \int _{0}^{\infty }\exp \left\{{\frac {1}{x}}\int _{0}^{x}\ln f(t )\,\mathrm {d} t\right\}\,\mathrm {d} x\leq \mathrm {e} \int _{0}^{\infty }f(x )\,\mathrm {d} x}
for any f ≥ 0.
Carleson's inequality [ edit ]
A generalisation, due to Lennart Carleson , states the following:[4]
for any convex function g with g (0) = 0, and for any -1 < p < ∞,
∫
0
∞
x
p
e
−
g
(
x
)
/
x
d
x
≤
e
p
+
1
∫
0
∞
x
p
e
−
g
′
(
x
)
d
x
.
{\displaystyle \int _{0}^{\infty }x^{p}\mathrm {e} ^{-g(x )/x}\,\mathrm {d} x\leq \mathrm {e} ^{p+1}\int _{0}^{\infty }x^{p}\mathrm {e} ^{-g'(x )}\,\mathrm {d} x.}
Carleman's inequality follows from the case p = 0.
An elementary proof is sketched below. From the inequality of arithmetic and geometric means applied to the numbers
1
⋅
a
1
,
2
⋅
a
2
,
…
,
n
⋅
a
n
{\displaystyle 1\cdot a_{1},2\cdot a_{2},\dots ,n\cdot a_{n}}
M
G
(
a
1
,
…
,
a
n
)
=
M
G
(
1
a
1
,
2
a
2
,
…
,
n
a
n
)
(
n
!
)
−
1
/
n
≤
M
A
(
1
a
1
,
2
a
2
,
…
,
n
a
n
)
(
n
!
)
−
1
/
n
{\displaystyle \mathrm {MG} (a_{1},\dots ,a_{n})=\mathrm {MG} (1a_{1},2a_{2},\dots ,na_{n})(n!)^{-1/n}\leq \mathrm {MA} (1a_{1},2a_{2},\dots ,na_{n})(n!)^{-1/n}}
where MG stands for geometric mean, and MA — for arithmetic mean. The Stirling-type inequality
n
!
≥
2
π
n
n
n
e
−
n
{\displaystyle n!\geq {\sqrt {2\pi n}}\,n^{n}\mathrm {e} ^{-n}}
applied to
n
+
1
{\displaystyle n+1}
implies
(
n
!
)
−
1
/
n
≤
e
n
+
1
{\displaystyle (n!)^{-1/n}\leq {\frac {\mathrm {e} }{n+1}}}
for all
n
≥
1.
{\displaystyle n\geq 1.}
Therefore,
M
G
(
a
1
,
…
,
a
n
)
≤
e
n
(
n
+
1
)
∑
1
≤
k
≤
n
k
a
k
,
{\displaystyle MG(a_{1},\dots ,a_{n})\leq {\frac {\mathrm {e} }{n(n+1)}}\,\sum _{1\leq k\leq n}ka_{k}\,,}
whence
∑
n
≥
1
M
G
(
a
1
,
…
,
a
n
)
≤
e
∑
k
≥
1
(
∑
n
≥
k
1
n
(
n
+
1
)
)
k
a
k
=
e
∑
k
≥
1
a
k
,
{\displaystyle \sum _{n\geq 1}MG(a_{1},\dots ,a_{n})\leq \,\mathrm {e} \,\sum _{k\geq 1}{\bigg (}\sum _{n\geq k}{\frac {1}{n(n+1)}}{\bigg )}\,ka_{k}=\,\mathrm {e} \,\sum _{k\geq 1}\,a_{k}\,,}
proving the inequality. Moreover, the inequality of arithmetic and geometric means of
n
{\displaystyle n}
non-negative numbers is known to be an equality if and only if all the numbers coincide, that is, in the present case, if and only if
a
k
=
C
/
k
{\displaystyle a_{k}=C/k}
for
k
=
1
,
…
,
n
{\displaystyle k=1,\dots ,n}
. As a consequence, Carleman's inequality is never an equality for a convergent series, unless all
a
n
{\displaystyle a_{n}}
vanish, just because the harmonic series is divergent.
One can also prove Carleman's inequality by starting with Hardy's inequality
∑
n
=
1
∞
(
a
1
+
a
2
+
⋯
+
a
n
n
)
p
≤
(
p
p
−
1
)
p
∑
n
=
1
∞
a
n
p
{\displaystyle \sum _{n=1}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{p}\leq \left({\frac {p}{p-1}}\right)^{p}\sum _{n=1}^{\infty }a_{n}^{p}}
for the non-negative numbers a 1 ,a 2 ,... and p > 1, replacing each a n with a 1/p n , and letting p → ∞.
Versions for specific sequences [ edit ]
Christian Axler and Mehdi Hassani investigated Carleman's inequality for the specific cases of
a
i
=
p
i
{\displaystyle a_{i}=p_{i}}
where
p
i
{\displaystyle p_{i}}
is the
i
{\displaystyle i}
th prime number. They also investigated the case where
a
i
=
1
p
i
{\displaystyle a_{i}={\frac {1}{p_{i}}}}
.[5] They found that if
a
i
=
p
i
{\displaystyle a_{i}=p_{i}}
one can replace
e
{\displaystyle e}
with
1
e
{\displaystyle {\frac {1}{e}}}
in Carleman's inequality, but that if
a
i
=
1
p
i
{\displaystyle a_{i}={\frac {1}{p_{i}}}}
then
e
{\displaystyle e}
remained the best possible constant.
^ T. Carleman, Sur les fonctions quasi-analytiques , Conférences faites au cinquième congres des mathématiciens Scandinaves, Helsinki (1923), 181-196.
^ Duncan, John; McGregor, Colin M. (2003). "Carleman's inequality". Amer. Math. Monthly . 110 (5 ): 424–431. doi :10.2307/3647829 . MR 2040885 .
^ Pečarić, Josip; Stolarsky, Kenneth B. (2001). "Carleman's inequality: history and new generalizations". Aequationes Mathematicae . 61 (1–2): 49–62. doi :10.1007/s000100050160 . MR 1820809 .
^ Carleson, L. (1954). "A proof of an inequality of Carleman" (PDF) . Proc. Amer. Math. Soc . 5 : 932–933. doi :10.1090/s0002-9939-1954-0065601-3 .
^ Christian Axler, Medhi Hassani. "Carleman's Inequality over prime numbers" (PDF) . Integers . 21, Article A53. Retrieved 13 November 2022 .
References [ edit ]
Hardy, G. H.; Littlewood J.E.; Pólya, G. (1952). Inequalities, 2nd ed . Cambridge University Press. ISBN 0-521-35880-9 .
Rassias, Thermistocles M., ed. (2000). Survey on classical inequalities . Kluwer Academic. ISBN 0-7923-6483-X .
Hörmander, Lars (1990). The analysis of linear partial differential operators I: distribution theory and Fourier analysis, 2nd ed . Springer. ISBN 3-540-52343-X .
External links [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Carleman%27s_inequality&oldid=1174905742 "
C a t e g o r i e s :
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H i d d e n c a t e g o r y :
● C S 1 : l o n g v o l u m e v a l u e
● T h i s p a g e w a s l a s t e d i t e d o n 1 1 S e p t e m b e r 2 0 2 3 , a t 1 3 : 3 1 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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● M o b i l e v i e w