Home » Posts tagged 'Inequality'

# 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,