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Tag Archives: Inequality
Let be positive real numbers such that . Prove that
Due to the AM – GM we have that
Let be a triangle and denote the lengths of the sides , and respectively. If then prove that
Applying Cauchy’s inequality to the vectors
we get that
due to the well known fact
It suffices to prove that . Let be two positive real numbers. Then,
On the other hand if then and
Then it follows that
Let such that . Prove that
It follows from Cauchy – Schwartz that
Fix . Apply the matrix on thus:
Since is exactly the – th column of the previous equality can be rerwritten as
Since this holds for all we get and the left inequality follows.
For a random unit vector the coordinate of the vector is . It follows from Cauchy – Schwartz that
Summing over all ‘s till we find we conclude that, for every unit vector , it holds that is less than the right hand side. Taking supremum with respect to all the right hand side inequality follows.
Let . Prove that:
Due to homogeneity we may assume . Thus there exist positive such that