Find the maximum of the value $c(n)$ similar to Hardy's inequality
Let $nge 2$ be give postive integer,and $a_{1},a_{2},cdots,a_{n}>0$,such $$a_{1}a_{2}cdots a_{n}=1$$
Find the maximum of the value $C(n)$ have
$$sum_{k=1}^{n}left(dfrac{1}{k}-dfrac{2}{n(n+1)}right)a_{k}ge C(n)left(sum_{k=1}^{n}dfrac{k^2}{a_{k}}right)^{frac{1}{n-1}}$$
This inequality is similar to Hardy's inequality ,Various proofs of Hardy's inequality
I have tried some methods but never solved it, such as using Cauchy-Schwarz inequality or induction method...
inequality contest-math
edited Nov 7 at 10:55
dmtri
1,3581521
asked Nov 1 at 8:00
inequality
767520
This question has an open bounty worth +100
reputation from inequality ending in 7 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
Let $nge 2$ be give postive integer,and $a_{1},a_{2},cdots,a_{n}>0$,such $$a_{1}a_{2}cdots a_{n}=1$$
Find the maximum of the value $C(n)$ have
$$sum_{k=1}^{n}left(dfrac{1}{k}-dfrac{2}{n(n+1)}right)a_{k}ge C(n)left(sum_{k=1}^{n}dfrac{k^2}{a_{k}}right)^{frac{1}{n-1}}$$
This inequality is similar to Hardy's inequality ,Various proofs of Hardy's inequality
I have tried some methods but never solved it, such as using Cauchy-Schwarz inequality or induction method...
inequality contest-math
edited Nov 7 at 10:55
dmtri
1,3581521
asked Nov 1 at 8:00
inequality
767520
This question has an open bounty worth +100
reputation from inequality ending in 7 days.
Looking for an answer drawing from credible and/or official sources.
A good way to prove this is to use the theorem 1.1 form this hkumath.hku.hk/~imr/IMRPreprintSeries/2006/IMR2006-32.pdf .
– max8128
51 mins ago
How to use this theroem 1? Thanks
– inequality
49 mins ago
add a comment |
Let $nge 2$ be give postive integer,and $a_{1},a_{2},cdots,a_{n}>0$,such $$a_{1}a_{2}cdots a_{n}=1$$
Find the maximum of the value $C(n)$ have
$$sum_{k=1}^{n}left(dfrac{1}{k}-dfrac{2}{n(n+1)}right)a_{k}ge C(n)left(sum_{k=1}^{n}dfrac{k^2}{a_{k}}right)^{frac{1}{n-1}}$$
This inequality is similar to Hardy's inequality ,Various proofs of Hardy's inequality
I have tried some methods but never solved it, such as using Cauchy-Schwarz inequality or induction method...
inequality contest-math
edited Nov 7 at 10:55
dmtri
1,3581521
asked Nov 1 at 8:00
inequality
767520
Let $nge 2$ be give postive integer,and $a_{1},a_{2},cdots,a_{n}>0$,such $$a_{1}a_{2}cdots a_{n}=1$$
Find the maximum of the value $C(n)$ have
$$sum_{k=1}^{n}left(dfrac{1}{k}-dfrac{2}{n(n+1)}right)a_{k}ge C(n)left(sum_{k=1}^{n}dfrac{k^2}{a_{k}}right)^{frac{1}{n-1}}$$
This inequality is similar to Hardy's inequality ,Various proofs of Hardy's inequality
I have tried some methods but never solved it, such as using Cauchy-Schwarz inequality or induction method...
inequality contest-math
inequality contest-math
edited Nov 7 at 10:55
dmtri
1,3581521
asked Nov 1 at 8:00
inequality
767520
edited Nov 7 at 10:55
dmtri
1,3581521
asked Nov 1 at 8:00
inequality
767520
edited Nov 7 at 10:55
dmtri
1,3581521
edited Nov 7 at 10:55
dmtri
1,3581521
edited Nov 7 at 10:55
dmtri
1,3581521
1,3581521
asked Nov 1 at 8:00
inequality
767520
asked Nov 1 at 8:00
inequality
767520
asked Nov 1 at 8:00
inequality
767520
767520
This question has an open bounty worth +100
reputation from inequality ending in 7 days.
Looking for an answer drawing from credible and/or official sources.
This question has an open bounty worth +100
reputation from inequality ending in 7 days.
Looking for an answer drawing from credible and/or official sources.
A good way to prove this is to use the theorem 1.1 form this hkumath.hku.hk/~imr/IMRPreprintSeries/2006/IMR2006-32.pdf .
– max8128
51 mins ago
How to use this theroem 1? Thanks
– inequality
49 mins ago
add a comment |
A good way to prove this is to use the theorem 1.1 form this hkumath.hku.hk/~imr/IMRPreprintSeries/2006/IMR2006-32.pdf .
– max8128
51 mins ago
How to use this theroem 1? Thanks
– inequality
49 mins ago
A good way to prove this is to use the theorem 1.1 form this hkumath.hku.hk/~imr/IMRPreprintSeries/2006/IMR2006-32.pdf .
– max8128
51 mins ago
A good way to prove this is to use the theorem 1.1 form this hkumath.hku.hk/~imr/IMRPreprintSeries/2006/IMR2006-32.pdf .
– max8128
51 mins ago
How to use this theroem 1? Thanks
– inequality
49 mins ago
How to use this theroem 1? Thanks
– inequality
49 mins ago
add a comment |
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A good way to prove this is to use the theorem 1.1 form this hkumath.hku.hk/~imr/IMRPreprintSeries/2006/IMR2006-32.pdf .
– max8128
51 mins ago
How to use this theroem 1? Thanks
– inequality
49 mins ago