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.

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

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 cauchy-schwarz

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

Siutsz, your substitution is very nice