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Tag Archives: Inequality

Root inequality

Let be positive real numbers such that . Prove that

Solution

Due to the AM – GM we have that

(1)

and

(2)

Thus,

Inequality of a triangle

Let be a triangle and denote the lengths of the sides , and respectively. If then prove that

Solution

Applying Cauchy’s inequality to the vectors

and

we get that

due to the well known fact

(1)

It suffices to prove that . Let be two positive real numbers. Then,

(2)

On the other hand if then and

(3)

Then it follows that

Root inequality

Let such that . Prove that

Solution

It follows from Cauchy – Schwartz that

Double inequality involving matrix

Prove that

Solution

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.

Homogeneity of inequality

Let . Prove that:

Solution

Due to homogeneity we may assume . Thus there exist positive such that

Hence,

Who is Tolaso?

Find out more at his Encyclopedia Page.