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2046
Joined: 02 Feb 2005Posts: 1

Posted: Mon Feb 14, 2005 4:03 pm Post subject: An inequality problem



Can someone help me?
Show that for and for a pair of real vectors and , the quantity
is bounded above by
For x=0 it is the Cauchy’s inequality and for x=1 it is an identity. 




Soarer
Frequent VisitorJoined: 18 Jan 2004Posts: 213
Location: Hong Kong

Posted: Mon Feb 14, 2005 5:16 pm Post subject:



My solution is very ugly. I think there is a better solution.
Then we immediately have some identities and inequalities.
Now,
The last inequality is true as
is the cauchy schwarz inequality and





Siutsz
Frequent VisitorJoined: 25 Dec 2003Posts: 37
Location: HK

Posted: Mon Feb 14, 2005 11:33 pm Post subject:



Here is another solution: The case x=0 is cauchy .
for , let for some real no. t
then the inequality becomes
which is equilvalent to
which follows from cauchyschwarz 




me
Frequent VisitorJoined: 12 Feb 2004Posts: 213
Location: Hong Kong

Posted: Tue Feb 15, 2005 12:42 am Post subject:



Siutsz, your substitution is very nice





